## Simplicity of supercooled liquids Experimental investigations inspired by the isomorph theory

Lisa Roed Schmidt, 2018. Supervisor: Kristine Niss, Co-Supervisor: Bo Jakobsen

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## EXP-potential and Quasi-universality

Andreas Kvist Bacher, 2018. Supervisor: Jeppe Dyre

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## Dynamics of glass-forming liquids: Will theory and experiment ever meet?

Henriette Wase Hansen, 2018. Supervisor: Kristine Niss, Co-Supervisor: Bernhard Frick

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The overall theme of this work has been to experimentally test the shoving
model and isomorph theory related to the dynamics of glass-forming liquids,
both of which, rather than being universal explanations, are expected to work
in the simplest case.
We test the connection between fast and slow dynamics in light of the shoving model from the temperature dependence of the mean-squared displacement from neutron scattering at nanosecond timescale and the elastic modulus from shear mechanics. We nd the fast dynamics to correlate with the alpha relax- ation time and fragility in agreement with predictions from the shoving model. The shoving model is tested on three liquids with simple dynamic behaviour in two versions, one formulated in terms of the instantaneous elastic modulus and one expressed in terms of the mean-squared displacement. We also test the underlying assumption connecting the two versions, directly relating the tem- perature dependence of the mean-squared displacement and that of the shear modulus. In the viscous liquid, we nd this to hold. We interpret the discrepancy at higher temperatures where the mean-squared displacement has a stronger temperature dependence than the shear modulus, as the alpha relaxation en- tering the neutron instrument window. In the view of the shoving model, the short-time properties govern the viscous slowing down.

We have developed a new sample cell for doing simultaneous dielectric and neu- tron spectroscopy at elevated pressure. This new high-pressure cell allows us to do experiments with high accuracy. From the dielectric signal, we can deter- mine the alpha relaxation time fast and with high precision in a large area of the temperature-pressure phase diagram while studying nano- and picosecond dynamics from neutron spectroscopy.

We use the new sample cell to locate isochrones, i.e. lines of constant alpha re- laxation time in temperature and pressure with the purpose of testing isomorph theory on three systems, two simple van der Waals and a hydrogen bonded liq- uid. We nd density scaling and isochronal superpositioning to hold for all three systems on alpha relaxation dynamics, and for the two van der Waals liquids, also when we have separation of timescales, i.e. the alpha relaxation is not contributing to the picosecond dynamics. The concept of isomorphs is observed to break down in two cases for the hydrogen bonding system: in density scal- ing of intramolecular motion and in isochronal superposition of the picosecond dynamics when there is separation of timescales. We show for one of the van der Waals liquids how the picosecond dynamics can be expressed as a function of the alpha relaxation time in agreement with the prediction of the existence of a one-dimensional phase diagram from isomorph theory, where one parameter is believed to control all dynamics.

## Structure and Dynamics of Hydrogen-Bonded Liquids. A Study of
Supra-Molecular Structures and Crystallisation

Mikkel
Hartmann Jensen, 2017. Supervisors: Kristine Niss, Tina
Hecksher, Christiane Alba-Simionesco

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This PhD thesis deals with the relastionship between structure
and dynamics of hydrogen-bonded liquids, focusing on the
effect of crystallisation on the dynamics, as well as the
dynamics of supra-molecular hydrogen-bonded structures, that
are believed to exist in these liquids. n-butanol and a series
of glycerol-water mixtures were used, because they are
hydrogen-bonded and can crystallise slowly, if the right
temperature protocol is followed.Two different sample cells were used to study the crystallisation of n-butanol with dielectric spectroscopy. The dielectric spectrum changed differently during crystallisation for the two cells. By modelling the Maxwell-Wagner effect, the differences could be explained by the sample cells giving rise to different morphologies of the crystal growth.

Neutron diffraction and small-angle scattering were used to study the pro- posed liquid-liquid transition in a glycerol-water mixture. The water in the mixture crystallised and no signs of a liquid-liquid transition were seen.

The dynamics of n-butanol was studied in the 120 K to 280 K range, using dielectric spectroscopy and neutron spin eco. The fitting procedure commonly used when analysing results from dielectric spectroscopy was shown to deliver unreliable results above 195 K for n-butanol.

Shear mechanical spectroscopy on glycerol revealed a slow relaxation process, which disappears gradually when water is added. This suggests that the slow relaxation process is connected to the hydrogen-bonded network between the glycerol molecules.

The above results stresses the need for combining different techniques when studying the dynamics of hydrogen-bonded systems. This is especially the case when studying crystallisation, where experiments should be done simultaneously to avoid influences from different sample cells.

## Isomorphs and pseudoisomorphs

Andreas Elmerdahl Olsen,
2016. Supervisor: Thomas Schrøder

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The present thesis deals with the scientific field of glass
and viscous liquids, and in particular the isomorphs theory,
through the application of computer simulations. The thesis is
divided into two main contributions. In the first part, the isomorph theory is applied to classical crystalline systems, in particular the single component Lennard-Jones fcc crystal, showing that even though the theory was developed for liquid systems, it works even better for crystalline systems. This is further confirmed by the investigation of six other model systems, two of which do not have isomorphs in the liquid phase; it is shown that systems without isomorphs in the liquid phase do not have isomorphs in the crystal phase either.

The second part of the thesis introduces the notion of pseudoisomorphs, characterizing systems without isomorphs but with isomorph-like behavior. These have been found in model systems with bonded interacts modeled as harmonic springs. Two methods for identifying pseudoisomorphic state points are developed and presented. The first relies on the fact that for the models used in this thesis, the eigenvalues of the potential energy Hessian fall into two distinct parts, one that corresponds to the harmonic bonds and one that cor- responds the remaining degrees of freedom. It is shown that the latter of these scale in accordance with the isomorph theory. The second method is of a more general nature, equating the reduced free energy of a system at different state points. Both methods leads to a number of implementations for identifying pseudoisomorphs. The methods are applied to two specific model systems, a simple asymmetric dumbbell model and a 10-bead Lennard-Jones chain model. Pseudoisomorphs are successfully found for both models.

## Isomorph theory and extensions

Lorenzo Costigliola,
2016. Supervisor: Jeppe Dyre

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This thesis has two main conceptual parts: in a first part the
isomorph theory is discussed and used to describe the
Lennard-Jones (LJ) system, following the path traced by
previous works from the Roskilde group, while in the second
part the theory is extended in different ways.Chap. 2 and the starting sections of Chap. 3 correspond to the first conceptual part. Here the isomorph theory is presented and then used to describe the freezing and melting invariances and the empirical Andrade equation for the freezing viscosity (which can also be thought as an invariance), showing that the isomorph theory can justify why these invariances hold and that they are not peculiarities of the freezing and melting processes.

At the same time Chap. 3 is also the start of the second conceptual part because the isomorph theory is used here for the first time to formulate predictions not along an isomorph but in its proximity, allowing for a precise description of the freezing and melting line. These two works on freezing and melting have been published in the first two companion papers. In the end of the same chapter the first part of an unpublished work is also presented. In this work a connection is suggested between the generalized LJ system starting to behave Roskilde simply and a change in the behavior of the scaling exponent

*γ*.

In Chap. 4 the isomorph theory is used to establish the temperature dependence of an isomorphic invariant quantity along an isochore, again 'going away from an isomorph' and investigating the connection between different isomorphs. This result has allowed the author to suggest a new equation describing the viscosity of the LJ system accurately in the whole Roskilde simple region.

Another extension of the theory is made by studying the LJ system (as prototype of Roskilde simple liquid) in 2d, 3d and 4d, verifying that the system conforms to hidden scale invariance (the symmetry consequence of isomorph theory) better in higher dimensions. These results have been published in the third companion paper and form the main matter of Chap. 5. In the same chapter the relation between the LJ system starting to behave Roskilde simple and the behavior of the scaling exponent is studied in 2d and 4d.

The thesis concludes with a work on a possible prototype non-simple system, the Gaussian core model. This study is presented in Chap. 6. The Gaussian core model exhibits strongly negative correlations and, for the first time, the existence of isomorphs with negative scaling exponent is reported, leading to a new application of the isomorph theory by including the Gaussian core system in the wide class of Roskilde simple systems.

## Tests of the isomorph theory by computer simulations of
atomic and molecular model liquids

Arnold Veldhorst,
2014. Supervisor: Thomas B. Schrøder

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Recently, it was discovered that some liquids behave in a
particularly simple way in part of their phase diagram
(Bailey et al., 2008a,b; Pedersen et al., 2008b). These
liquids can be identified in computer simulation by
correlations between their potential energy and virial,
which is quantified by the correlation coefficient R being
higher than 0.9. Their simplicity lies in the fact that they
have curves in their phase diagram called isomorphs, along
which many properties are invariant to a high degree. This
includes for example constant configurational entropy,
constant isochoric specific heat, invariant equilibrium
dynamics, and invariant equilibrium structure (Gnan et al.,
2009; Schrøder et al., 2009). Whereas previous studies have
focused mainly on atomic model liquids and small rigid
molecules (Ingebrigsten et al., 2012a), this thesis aims to
show that the isomorph theory is applicable to a much wider
range of systems. We show that colloidal suspensions and dusty plasmas may also be considered simple by simulating the Yukawa potential. The Yukawa fluid is shown to have correlated energy-virial fluctuations and isomorphs in its phase diagram. We show that it is possible to predict the shape of the isomorph from the form of the potential. The shape of an isomorph in the density-temperature phase diagram is described by a function

*h(ρ)*. Recent results for real liquids show that the logarithmic slope

*γ*= d ln

*h(ρ)*/d ln

*ρ*increases with density (Bøhling et al., 2012). We show with computer simulations of two novel potentials and the Girifalco potential that this may be an effect of the finite molecular volume of many organic glass formers.

Before the development isomorph theory it was already found that for many real liquids, the dynamics can be scaled onto a single curve by a power law of density. A large part of the real liquids that have been shown to obey this scaling are polymers (Roland et al., 2005). For atomic models and small rigid molecules this scaling has been shown to be an approximation of isomorph theory in small density ranges (Gnan et al., 2009; Ingebrigtsen et al., 2012). We simulated exible chains of Lennard-Jones particles connected by rigid bonds, and show that this liquid has correlations and isomorphs. Interestingly, we find that the structure of a single molecule is not invariant on the isomorph, while the interatomic structure is. Both the dynamics of the Lennard-Jones particles, as well as the dynamics of the entire chain are found to be invariant on the isomorph. Our results indicate that isomorph theory may also hold for liquids consisting of long, flexible chains, like polymers.

Simulations of Lennard-Jones chains where the covalent bonds are simulated as harmonic springs has led to the discovery of so-called pseudo isomorphs. The inclusion of these exible bond destroys the energy-virial correlations almost completely (R≈0.28). We show that these chains nevertheless have curves in their phase diagram along which the dynamics and the intermolecular structure is invariant. We show that the energy-virial correlations do persist in the frequency domain at low frequencies.

## From Isomorphs in Molecular Liquids and Confinement to
Molecular Dynamics at Constant Potential Energy

Trond
S. Ingebrigtsen, 2013. Supervisor: Jeppe C. Dyre

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Strongly correlating liquids are defined by having strong
correlations in the *NVT*ensemble between the equilibrium fluctuations of the potential energy

*U*and the virial

*W*. These liquids were discovered by researchers of the ''Glass and Time'' group, Roskilde University, via molecular dynamics computer simulations and detailed in a series of papers. Strongly correlating liquids have also been verified in experiments. This thesis expands on the basic understanding of strongly correlating liquids in three directions and uses motivation derived from these liquids to perform research in a fourth direction.

(1) We show that strongly correlating liquids have a rather simple thermodynamics in the sense that temperature separates into a product of a function of excess entropy per particle and a function of density. This fact leads to the proposal of a more general scaling procedure, the ''isomorph scaling'', which is in contrast to the traditional ''density scaling'' procedure that breaks down when considering larger density changes than usually applied in experiments. In addition, we show that the expressions of Rosenfeld and Tarazona for the potential energy and heat capacity along an isochore holds to a better approximation for strongly than non-strongly correlating liquids.

(2) Strongly correlating liquids are characterized by having isomorphs to a good approximation. Isomorphs are curves in a liquid's phase diagram along which structure and dynamics are invariant in reduced units as well as some thermodynamic quantities. Originally, isomorphs were investigated for bulk atomic systems. We extend the concept of isomorphs to systems composed of rigid molecules and show that these systems can have isomorphs to a good approximation, too. We also extend the isomorph concept to confined liquids. Confined liquids exhibit stratification, i.e. the particles of the liquid order themselves in well-defined layers, and position-dependent relaxation processes. Despite of these facts, confined liquids have isomorphs to a good approximation. This observation establishes a connection to a novel excess entropy scaling procedure for predicting confined-liquid behavior via knowledge of their bulk relationships.

(3) We propose that strongly correlating liquids are to be identified with simple liquids. This new definition of ''what is a simple liquid?'' is in contrast to the more traditional definition as being systems with radially symmetric pair potentials. The motivation for this new definition is derived from a discovery relating to strongly correlating liquids, namely that structure and dynamics are determined to a good approximation

*only*by the interactions within the first coordination shell (FCS), i.e. the nearest-neighbor interactions. In fact, we show that the FCS property holds to a good approximation also in confinement. Bulk and confined liquids are thus more closely related than what is traditionally believed to be the case.

(4) We present a new molecular dynamics,

*NVU*dynamics that, instead of conserving the energy

*E*as in standard Newtonian

*NVE*dynamics, conserves the total potential energy

*U*. From simulations and theoretical arguments, we show that

*NVE*and

*NVU*dynamics become equivalent in the thermodynamic limit for both atomic and molecular systems.

*NVE*and

*NVU*dynamics may thus be used interchangeably in simulations for most purposes.

## Computer simulations of viscous liquids and aspherical
particles

Lasse Bøhling, 2013. Supervisor: Thomas
B. Schrøder

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This Ph.D. thesis falls into two basic parts. Part 1 explores a
class of liquids commonly referred to as *Strongly Correlating Liquids*. The central aim of the first part of the study is to establish a better understanding of these liquids through an investigation of their pair interaction potential. Part 2 examines the socalled

*rolypoly-particle*, an aspherical surface of constant width. This part of the study focuses on the self assembly and densest packing of this particle. The overall objective of this part of the study is to connect the shape of the particle directly to its glass forming ability. Both parts rely on theoretical observations combined with

*Molecular Dynamics*and

*Monte Carlo*simulations.

The main conlusions are as follows:

For liquids with strong virial and potential energy fluctuations in the canonical ensemble, the two dimensional (density and temperature

*(ρ,T)*) phase diagram can be reduced to one variable

*(h(ρ)/T)*. The scaling function

*h(ρ)*is derived analytically for atoms interacting via a pair potential constituting a sum of Inverse Power Laws:

*φ(r) = Σ*. It is shown how the scaling function

_{n}ε_{n}(σ_{n}/r)^{n}*h(ρ)*directly links to the pair interaction potential.

The glass forming ability of the rolypoly's is to a first approximation determined by the non-sphericity of the particle and for high pressures, crystallization is controlled by diffusion, consistent with classical nucleation theory. Densest packing is found for the Rolypoly with a packing fraction ≈ 0.7698, having two particles in the unit cell.

## Testing predictions of the isomorph theory by experiment

Ditte Gundermann, 2013. Supervisor: Kristine Niss

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The overall theme of the present PhD thesis is an experimental
approach to the concept of strongly correlating liquids and
isomorphs. Three theoretical predictions are treated:1) The identity between the density scaling exponent γ

_{scale}and the fluctuation exponent γ

_{isom}(chapter 6). It is shown how the fluctuation exponent can be determined through linear-response measurements. By such measurements on a van der Waals bonded liquid, tetramethyl-tetraphenyl-trisiloxane (DC704), the resulting value of γ

_{isom}is compared to the value of γ

_{scale}found from high-pressure dielectric measurements. The results show that the two values for the exponent agrees well, which implies that strongly correlating liquids exist. These liquids can be viewed as a class of simple liquids.

2) The state-point dependence of the density-scaling exponent (chapter 7). Data sets for two liquids, DC704 and polyphenylether (5PPE), are examined with respect to the temperature and density dependence of the density-scaling exponent. The density range for the data turns out to be too small to draw a conclusion on the matter.

3) The aging properties of strongly correlating liquids (chapter 8). An idea for testing the predicted and in computer simulations observed aging properties of strongly correlating liquids is presented (chapter 8), along with some experimental challenges involved with the experiment. The experiment is to be carried out in a new setup for high-pressure dielectric spectroscopy. A detailed description of the setup is given and the main challenges involved with the implementation, in particular in relation to the proposed experiment, are described.

Furhtermore, a set of data on the frequency dependent adiabatic bulk modulus for two molecular liquids, trimethyl-pentaphenyl-trisiloxane (DC705) and dibuthyl phtalate (DBP), are presented and analyzed in terms of relaxation time, time-temperaturesuperposition (TTS) and high-frequency slope of the alpha relaxation (chapter 3). The bulk modulus measurements are one of the four frequency-dependent response functions used to calculate γ

_{isom}.

## Pilot experiment to measure a complete triple of
thermoviscoelastic response functions of equilibrium viscous
liquids and an effect of thermoelastic coupling - cooling by
heating

Jon Papini, 2011. Supervisor: Tage Christensen

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The present PhD thesis dissertation is a report on two
main projects: (1) A study of thermoviscoelastic phenomena, and
(2) a pilot experiment to test if relaxation of highly viscous
liquids can be described by a single internal parameter.
The first part is a study of a peculiar thermoelastic effect - cooling by heating: If a solid sphere is heated at the surface, temperature will drop in the middle of the sphere. The study gives a theoretical explanation in terms of an analysis of the full thermoviscoelastic problem in spherical geometry. The effect is shown to be the consequence of a non-trivial thermomechanical coupling which exist if there is a difference between the longitudinal and the isobaric specific heat. Through extensive mathematical modeling I derive a formula that predicts the size of the cooling by heating effect when a finite amount of heat is added at the surface of a thermoelastic solid. Numerical simulations prove that the effect also exists in the thermoviscoelastic case, and it is shown that the effect is present in the liquid, even in the case where the effect is non-present in the solid. The modeling also shows that the effect is not limited to the center of the sphere and thus a detection of the phenomenon is not dependent on a very precise central placement of a thermometer. Besides the theoretical work I have done molecular dynamics simulations of a nano-sized droplet proving that cooling by heating is present also on the molecular level. Finally, I have performed measurements in the lab on a sphere of glucose. The experiments show that the effect is present also in real systems.

The second part is a pilot experiment. I explore the possibility of measuring a complete set of thermoviscoelastic response functions on the same sample under identical conditions, by combining two devices used and developed in the Glass & Time group at Roskilde University. The motive for this is the scientific question whether one or more internal parameters are needed to describe the relaxation dynamics of highly viscous liquids. This question is a part of the long term research plan of the group. The first device is the Piezoelectric Bulk modulus Gauge (PBG), which is a spherical piezoelectric shell coated with electrodes on both sides. Applying an oscillation electric field to the liquid filled PBG one measures the mechanical auto response function K

_{S}- the adiabatic bulk modulus. The second device is a thermistor. Employing the 3ω-method one measures the thermal auto response function c

_{l}- the longitudinal specific heat. Placing the thermistor in the center of the PBG, one can perform the third experiment: The measurement of the cross response function β

_{S}= (∂V/∂S)

_{p}- the adiabatic pressure coefficient. Here a heat current is generated in the center with the thermistor while the resulting deformation of the PBG as the liquid expands is measured as a piezoelectric voltage. This is a novel measurement never done before, and in order to deduce β

_{S}one first need to measure the two auto response functions. This part of the thesis briefly presents the measurements of K

_{S}and c

_{l}. Also I show how to model the third combined experiment, and perform a first pilot measurement of the cross response function. The measurement gives a good signal to noise ratio, but the deduced spectrum found for β

_{S}has peculiar characteristics. The absolute value of the measured property is in the correct range, but the temperature dependence of the dispersion is not understood. It is suggested that the model used is to simple and that the thermal structure of the thermistor bead needs to be included in the model. A test of “single parameterness” of relaxation in an equilibrium liquids is not possible due to the quality of the data.

## Relaxation in supercooled liquids. Linear and Nonlinear,
Mechanical and Dielectric Studies of Molecular Liquids.

Tina Hecksher, 2010. Supervisor: Jeppe C. Dyre

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In the present PhD thesis dissertation three themes of viscous
liquids dynamics are treated: (1) the visco-elastic properties
studied via mechanical measurements, (2) the temperature
dependence of the (dielectric) relaxation time, and (3)
non-linear relaxation. The first part is based on own
measurements of molecular liquids, in particular
5-phenyl-4-ether (PPE) and tetramethyl-tetraphenyl-trisiloxane
(DC704), while part 2 and 3 are based on analyses of
measurements by others.The mechanical measurements in the first part consist of bulk and shear modulus measurements with unique methods developed in the Glass & Time group at Roskilde University. The measurements were carried out under identical experimental conditions in the same cryostat thus eliminating most uncertainties associated with comparing different measured properties of viscous liquids, the temperature calibration being the most problematic. We found that the spectral shape of the bulk and shear modulus relaxation is identical within experimental uncertainty for PPE and DC704, respectively, while the bulk modulus relaxes slower than the shear modulus. Bulk and shear modulus relaxation times, as well as relaxation times of three other response functions, are however proportional in the entire temperature interval studied.

Measurements on DC704 were combined with mechanical measurements at higher frequencies carried out by our collaborators from Massachusetts Institute of Technology in the USA to compile “the widest mechanical spectrum in the world”. A total of six different techniques covers 14 orders of magnitude in frequency and 250 Kelvin in temperature. The good agreement between the results from all these methods can be seen as a mutual confirmation of the absolute values measured by the individual methods.

In the second part Adam-Gibbs entropy model is challenged. The model predicts a phase transition to an “ideal glassy state” with infinite relaxation time. An analysis of the temperature dependence of the dielectric relaxation time for 42 liquids show that data do not favor this picture.

In the third part, a series of non-linear relaxation measurements on five different molecular liquids. It is shown that the non-linear relaxation can be linearized via the Tool-Narayanaswamy formalism. The high resolution of the measurements indicates that the linearized relaxation at long times follow a simple exponential, while the relaxation at short times is “stretched”. This discovery lead to a new single-parameter fitting function for the relaxation of viscous liquids. The function was fitted to the (linear) dielectric relaxation spectra for 53 liquids and it was found to fit as well as the “standard” fitting function, the stretched exponential.

## Viscous liquid dynamics in and out of equilibrium. From
isomorphic curves in the phase diagram to effective
temperatures in ageing.

Nicoletta Gnan, 2010. Supervisor:
Thomas B. Schrøder

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This doctoral thesis has eight companion papers.The main part of the work is based on the study of strongly correlating liquids and isomorphs. It is shown the existence of a class of liquids for which it has been observed a strong correlation between the fluctuations of the potential energy and of the virial at constant volume. This class of liquids is called “strongly correlating liquids” (SCLs). These liquids are characterized by two important numbers: the coefficient

*R*that quantifies the correlation (strong correlation corresponds to

*R*>0.9) and $\gamma$ that corresponds to the slope resulting from plotting the fluctuations of the virial against those of the potential energy.

A perfect correlation is trivially found in soft-spheres whose potential is an inverse power-law (IPL); in such case the potential energy is proportional to the virial through a factor given by

*γ=n*

_{IPL}/3 where

*n*

_{IPL}is the exponent of the inverse power-law. For other SCLs instead an explanation of the strong correlation is given if the pair potential can be approximated with an

*extended inverse power-law*(eIPL), i.e. an inverse power-law plus a linear term. In this approximation, the correlation is explained by the fact that only the inverse power-law term contributes to the fluctuations and therefore to strong correlation.

In this work it is shown that SCLs inherits to good approximation a number of properties from IPLs. In particular simulations show that SCLs surprisingly inherit scale invariance, i.e. state points with the same

*ρ*have a number of thermodynamic, dynamic and structural properties that are the same in scaled units. The curves identified by the relation above are characterized by state points having proportional Boltzmann factors and are called “isomorphs”. An extended theoretical and numerical investigation has been done to test the isomorph properties which can be found not only at equilibrium but even in the aging regime. The equations of isomorphs for generalized Lennard-Jones systems are derived and for this case it is shown that the shape of isomorphs depends only on the exponents of the potential and not on its parameters.

^{ γ}/TTo investigate the properties of isomorphs in the glassy state the fluctuation dissipation relation (FDR) is used showing that the glassy line defined by the effective temperature calculated via the FDR is an isomorph. Moreover it is shown that from a single aging experiment (i.e. from a temperature/density jump) it is possible to predict with a very good approximation the effective temperatures of all the aging experiments that can be performed.

Part of the thesis is dedicated to the energy landscape approach for the study of supercooled liquids. In particular the attention is focused on the evolution in time of the probability distribution function (PDF) of inherent states (IS) energy during temperature jumps from low to high temperatures. It is shown that small systems exhibit a double peak distribution of metabasins energy in analogy with the trap model. For large systems instead this effect is smeared out and the time evolution of the PDF suggests that the big system can be thought as a superposition of weakly interacting subsystems having - on average - the same mean value and the same variance of the PDF of the large system.

## Experimental Studies of Supercooled Liquids, Gels and
Glasses. Dynamic Correlation Lengths and Off-Equilibrium
Dynamics.

Claudio Maggi, 2010. Supervisor: Jeppe C. Dyre

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This PhD thesis is accompanied by a number of papers whose main
focus is the investigation of the dynamics of supercooled
liquids, gels and glasses. Moreover new theoretical and
experimental results are presented. Two of the articles presented report frequency-resolved (broad-band) shear-mechanical measurements acquired via a unique technique developed by our group (the piezo-shear-gauge (PSG) method). In one of these we measure the dynamic shear modulus of five glass-forming liquids close to the glass transition temperature. The analysis includes a characterization of the temperature-evolution of the relaxation time of the mechanical response and of its shape. The measurements of the dynamic shear modulus allow for a test of the shoving model that links the structural relaxation time to the infinite-frequency shear-modulus. In the second work employing the PSG method the dynamic shear-modulus of two monohydroxy molecular liquids (monoalcohols) is determined. These measurement are compared with dielectric-spectroscopy measurements performed on the same liquids and in the same experimental conditions. The findings of this work indicate that the additional Debye-process (that dominates the dielectric response of these monoalcohols) can not be detected in the frequency-resolved shear-modulus. This mechanical response function displays indeed only a clear structural relaxation in agreement with dynamic light scattering and calorimetric measurements. These results are illustrated and expanded in Chapter 2.

The off-equilibrium dynamics of gels and glasses was also subject of investigations as it can be found in the other companion articles. These studies have been carried by means of experiments and computer simulations. An off-equilibrium gel-forming colloidal solution is studied combining depolarized dynamic light scattering measurements and Kerr-effect response measurements. The combined use of these two measurements allow one to determine whether the fluctuation-dissipation theorem is violated out of equilibrium and what is the effective temperature associated with this non-equilibrium condition. The findings of this work are presented in Chapter 3. A computer simulation study of the violation of theorem is also presented. This is done for a widely studied a model glass-former taken out of equilibrium via fast cooling and/or densification. It is shown how the special properties of this liquid (shared by the entire class of strongly correlating liquids) imply that the effective temperature depends only on the final density of the off-equilibrium jump.

In addition it is shown how only for these liquids a single off-equilibrium simulation allows one to predict the effective temperature of any glass produced by an arbitrary density/temperature jump. This work is illustrated and expanded by new findings in Chapter 4.

The dynamic heterogeneity of the glassy dynamics in and out of equilibrium was studied as a further subject. Additional shear-modulus measurements were studied extracting the characteristic dynamic correlation volumes associated with this mechanical relaxation. A separate paper was dedicated to this analysis. The estimation of the four-point correlator reported in this work is based on some approximation schemes that allow one to quantify this via a more accessible three-point susceptibility A careful comparison of this quantity obtained by means of dielectric measurements reveals that the number of dynamically correlated molecules estimated is generally slightly different for the shear and dielectric response. Nevertheless the analysis indicates that these characteristic correlation volumes grow proportionally as the temperature is lowered. These results are presented in Chapter 5 where we also illustrate a new experiment for the direct measurement of the four-point susceptibility obtained by combined dynamic light scattering techniques. In Chapter 6 we present a new model for the connection of the shear and the dielectric response accounting for the presence of the dynamic heterogeneities.

## The primary relaxation in glass-forming liquids : an
empirical investigation of dielectric data

Albena Ivanova
Nielsen, 2009. Supervisor: Jeppe C. Dyre

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The focus in this study is on the high frequency decay of the
relaxation in terms of the “minimum slope” of the
α dielectric loss vs. the loss peak frequency in log-log
plot. The analysis is kept model-independent in an attempt to
perform an unbiased conclusion.The dielectric data collection
consist of measurement for 53 liquids at ambient pressure and
high pressure (isothermal conditions) data sets for 10
liquids. The data are provided from different groups and
supplemented by new measurements. The main conclusion is: the
most frequently observed minimum slope is close to −1/2,
corresponding to approximate √t dependence of the
dielectric relaxation function at short times. The study
encloses an investigation of possible correlations between
minimum slopes and: 1) Temperature quantified via the loss-peak
frequency; 2) How well an inverse power law fits data above the
loss peak; 3) Degree of time-temperature superposition for data
at ambient pressure; 4) Loss-peak half width, and stretching
exponent; 5) The phenomenological classification in A and B type
liquids; 6) Deviation from non-Arrhenius behavior; 7) Loss
strength; 8) Temperature pressure co-invariance of the spectral
shape along isochrones (pressure-temperature superposition the
same relaxation time) . The result is: minimum slopes close to
−1/2 correlates to the listed in the first three and the
very last points, which indicates a special status of liquids
with minimum slopes close to −1/2. Concerning the last
points only fairly insignificant correlations are found with the
exception of large-loss liquids, which have minimum slopes that
are numerically significantly larger than 1/2 and loss peak
widths that are significantly smaller than those of most other
liquids. We conclude that – excluding largeloss liquids
– approximate √t relaxation appears to be a generic
property of the α relaxation of organic glass
formers. There are some secondary conclusions: The large loss
liquids are typically A-type. Besides this purely descriptive
classification in two types do not give some new insight, since
there is no correlation to other than the shape parameters of
the relaxation process. The high pressure study shows that the
model independent minimum slope as a shape parameter captures
phenomena that are reported for the different liquids like the
pressure-temperature superposition. There is indication that it
captures hidden beta processes.

## Long-time simulations of viscous liquids: from strong
correlations to crystallization

Ulf R. Pedersen,
2009. Supervisor: Thomas B. Schrøder

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This philosophiæ doctor thesis has thirteen companion
papers. The major part is a theoretical and simulation study of
a class of liquids referred to as strongly correlating
liquids. However, one chapter is dedicated to a study of
thermodynamic fluctuations of simulated phospholipid membranes,
and another to a simulation study of crystallization of a binary
mixture. The thesis also includes a short introduction to
supercooled viscous liquids and molecular dynamics
simulations. The final chapter is an outlook. The conclusions
are as follows:A class of model liquids exhibit strong correlation in the thermal fluctuations of the virial and potential energy at constant volume. The origin of the correlation is explained for the Lennard-Jones liquid. The pair energy can to a good approximation be replaced by an inverse power-law plus a linear term. At constant volume only the inverse power-law gives a contribution to fluctuations, thereby explaining the correlation. Two quantities characterize a strongly correlating liquid: i) A correlation coefficient, R, that determine the degree of correlation and ii) a slope,

*γ*, that determines the proportionality between virial and potential energy. If the correlation coefficient is close to unity the liquid is strongly correlating and 3

*γ*equals the exponent of the inverse power-law.

For viscous liquids, the correlation coefficient equals the inverse square-root of a Prigogine-Defay ratio defined from three response functions. Also the slope can be determined from response functions. Literature values of the classical Prigogine-Defay suggests that van der Waals bonded liquids in general are strongly correlating.

Strongly correlating viscous liquids are more simple than viscous liquids in general:

i) Slow dynamics are determined by a single relaxing parameter. Thus, time- and frequency-dependent response functions of a given ensemble are proportional. ii) Scale invariance is inherited from soft-sphere liquids though the effective inverse-power law. Thus, state points with the same value of

*ρ*have the same scaled structure and dynamics. These conclusions are verified in simulations.

^{ γ}/TSlow thermal fluctuations of volume and energy of simulated phospholipid membranes are strongly correlated. The origin of the strong correlation can be traced to the van der Waals bonded core of the membranes and, thus, have the same origin as simple strongly correlating liquids.

The last part of the thesis reports crystallization into the MgZn

_{2}Laves phase of the binary Lennard-Jones mixture suggested by Wahnström [1991]. The crystallization mechanism from the supercooled melt is investigated in detail.

## Energy Landscape Approaches to the Dynamics of supercooled
liquids

Niels L. Ellegaard, 2005. Supervisor: Jeppe C.
Dyre.

My thesis consists of two separate halves. In the first half I
propose a new test for determining whether or not the combined
relaxation of volume and enthalpy in a supercooled liquid is
described by a single parameter model. The single parameter
hypothesis has been rejected experimentally for a number of
systems, but data on supercooled molecular liquids is very
sparse. Furthermore most of the results rely on extrapolations
from non-linear experiments. On the contrary recent computer
simulations suggest that the linear response of a disordered
systems of Lennard spheres is well described by a a single
parameter model. My test requires the measurement of three
complex responses functions at a single frequency. In other
words the experimentalist can completely avoid the problems
related to measuring instantaneous (infinite frequency)
response. I present a master equation formalism for calculating
isotropic thermoviscoelastic response functions, and I prove
that a system is described by a single parameter model if there
is a direct correlation of the volumes and “inherent
enthalpies” of the local minima of the potential energy
function.In the second half I examine stationary configurations of a simple model potential (Lennard-Jones). I report two unsuccesful attempt device a method for constructing 2

^{m}of stationary configurations of different saddle order, in such a way that they correspond to the corners of a

*m*-dimensional parallelepiped. By examining the energies of these configurations I wanted to define an energy barrier, and according to recent work by Angelani this energy barrier should be related to the dynamics of the system just above the critical mode coupling temperature. My first attempt on parallelepipeds is based on the one dimensional manifold of “gradient extremals”, whereas the second attempt is based on the streamlines of the “image reversal” vector field. However both definitions lead to unphysical neighboring relations.

## Frequency-dependent heat capacity - Experimental work to
improve and understand planar heater experiments using the
3-omega detection technique

Claus F. Behrens,
2004. Supervisor: Tage Christensen.

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The frequency dependent heat capacity of super-cooled glycerol near the glass-transition is measured using the 3-omega detection technique. An electrical conducting film with a temperature-dependent electrical resistance is deposited on a substrate. The thin film is used simultaneously as a heater and a thermometer. The aim of the work is to improve and understand this planar heater experiment. I find:

- Carbon has advantages as heater material over the traditionally used metal (nickel) heaters.
- The thermal coupling to the surrounding temperature bath should not be made through the liquid, but through the substrate.
- Edge effects, as a result of the finite size of the heater, play an important role. The traditional way of dealing with these effects are not entirely correct.
- The Cole-Davidson function with Beta_CD = 0.62 +- 0.02 fits the measured data for glycerol

## Hopping in Disordered Media: A Model Glass Former and a
Hopping Model

Thomas Schrøder, 1999. Supervisor: Jeppe C.
Dyre.
cond-mat/0005127

Two models involving “hopping” are studied:A model glass-forming liquid is investigated by molecular dynamics under (pseudo-) equilibrium conditions. ``Standard'' results such as pair-correlation functions, mean square displacements, intermediate scattering functions, etc. are reported. At low temperatures hopping is present in the system as indicated by a secondary peak in the van Hove self correlation function. The dynamics of the model is analyzed in terms of its potential energy landscape, and we present direct numerical evidence for a 30 years old picture of the dynamics at sufficiently low temperatures. The fundamental transitions in the dynamics are found to involve particles moving in a cooperative string-like manner.

The symmetric hopping model is analyzed in the extreme disorder limit (low temperatures) using the Velocity Auto Correlation (VAC) method. The VAC method is derived in this thesis and has the advantage over previous methods, that it can calculate a diffusive regime in finite samples. Numerical results using the VAC method are compared to three analytical approximations, including the Diffusion Cluster Approximation (DCA), which is a new (and so far unpublished) approximation developed by my supervisor Jeppe C. Dyre.

## Fluctuation and Linear Response in Supercooled Liquids

Johannes K. Nielsen, 1999. Supervisor: Jeppe C. Dyre.

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Fluctuation dissipation theorems are derived for thermodynamic
properties like frequency dependent specific heat and
compressibility. First the case where the dynamics is
restricted by constant volume and energy is considered. The
dynamical linear response to a heat pulse and a volume change
at time zero is calculated, under assumption of energy
conservation. Then the case of isothermal isobaric conditions
are treated by a slight modification of ordinary linear
response theory. In both cases the perturbation cannot be
stated through the Hamiltonian, but has to be imposed by
variation of the external thermodynamic system parameters. In
thermodynamic response theory equivalence between ensembles is
broken, but time correlation functions sampled in different
ensembles are connected through the Maxwell relations of
thermodynamics generalized to the frequency domain. Different
applications of the theory in the field of supercooled liquids
are demonstrated. First the full frequency dependent
thermodynamic response matrix is extracted from simulations of
a binary Lennard-Jones liquid. Secondly, some simple
stochastic models of supercooled liquids are analyzed in the
framework of linear thermodynamic response theory. In addition
low temperature universality of the specific heat is
discussed.Analysis of hydrogen bond dynamics in supercooled SPC/E model water shows that there is a separation between a fast (local) time scale, and a slow (collective) time scale in the supercooled regime. Time temperature scaling of the hydrogen bond correlation function is discussed in terms of a diffusion model.