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Neutron Scattering Applications and Techniques
Series Editors:
Ian S. Anderson
Neutron Sciences Directorate
Oak Ridge National Laboratory
Oak Ridge, TN 37831-6477
USA
[email protected]
Alan J. Hurd
Lujan Neutron Scattering Center at LANSCE
Los Alamos National Laboratory
PO Box 1663, MS H805
Los Alamos, NM 87545
USA
[email protected]
Robert L. McGreevy
Neutron Sciences Directorate
Oak Ridge National Laboratory
Oak Ridge, TN 37831-6477
USA
[email protected]
For further volumes:
http://www.springer.com/series/8141

Victoria Garc´ıa Sakai • Christiane Alba-Simionesco
Sow-Hsin Chen
Editors
DynamicsofSoftMatter
Neutron Applications
123

Editors
Victoria Garc´ıa Sakai
CCLRC Rutherford Appleton Laboratory
ISIS Facility
Chilton
OX11 0QX Didcot, Oxon
United Kingdom
[email protected]
Sow-Hsin Chen
Department of Nuclear Science
and Engineering
Massachusetts Institute of Technology
Massachusetts Avenue
77 Cambridge, Massachusetts 02139
USA
[email protected]
Christiane Alba-Simionesco
Laboratoire L´eon Brillouin
UMR 12 CEA-CNRS
CEA Saclay, Gif sur Yvette
Bˆatiment 563
France
[email protected]
ISSN 1868-0372 e-ISSN 1868-0380
ISBN 978-1-4614-0726-3 e-ISBN 978-1-4614-0727-0
DOI 10.1007/978-1-4614-0727-0
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011943091
© Springer Science+Business Media, LLC 2012
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
connection with any form of information storage and retrieval, electronic adaptation, computer software,
or by similar or dissimilar methodology nowknown or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are
not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject
to proprietary rights.
Cover illustration:cRowan Hargreaves, ISIS Facility, UK
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)

Preface
The broad ﬁeld of Soft Matter (polymers, proteins, colloids, liquid crystals, and
so on) has experienced an explosive growth in the last few decades and has never
been more vibrant. The advances in technology and experimental methods, in theory
and simulations, and the search for new “smart” materials to address social and
global challenges, continues to expand the research in this ﬁeld. The properties of
Soft Matter systems lie across many disciplines – physics, biology, engineering,
and chemistry – and the cross talk between scientists in all ﬁelds is of uttermost
importance to gain a complete understanding of these systems. In addition, their
characteristics span over a wide range of lengthscales and timescales which requires
the combination of theoretical and simulation techniques, with a number of different
experimental techniques.
The importance of neutron scattering techniques was conﬁrmed with the Nobel
Prize award in 1994 to Shull for “. . . the development of the neutron diffraction
technique...”andBrockhouse for “. . . for the development of neutron spectroscopy
....” Neutrons tell us “where atoms are and how they move.” Three years earlier
Pierre-Gilles de Gennes, a pioneer in polymer physics, was also awarded the Nobel
Prize for “. . . discovering that methods developed for studying order phenomena in
simple systems can be generalized to more complex forms of matter, in particular
to liquid crystals and polymers.” In his Nobel lecture on Soft Matter he addressed
their two most important properties,complexityandﬂexibility. Neutron scattering
is thus an ideal candidate for the characterization of soft matter systems. In fact, it
was data from small angle neutron scattering measurements thatprovided evidence
supporting the work of another Nobel Laureate, Paul Flory, in his prediction that
polymer chains adopt self-avoiding random walks.
Structural characterization of Soft Matter systems is the initial step to under-
standing such materials, but ultimately many of their macroscopic properties such as
viscosity, conductivity, or enzymatic function, are directly related to their molecular
motions. Thus, it is necessary to obtain a dynamical characterization as well.
Neutrons offer the advantage that they provide temporal and spatial information
simultaneously and especially for Soft Matter systems, are able to discriminate
between H and D isotopes, allowing component selectivity in experiments. This
v

vi Preface
is of particular relevance to understand for example, the functionality of proteins in
the presence/absence of water, the conﬁnement of water in soft micellar systems, the
mixing of two polymers, or the preparation of polymer nanocomposites to achieve
new materials with tuneable properties.
The aim of this book is to provide scientists, engineers, and advanced stu-
dents with a reference on how neutrons are a key tool for the study of the
dynamical processes in soft materials. It also hopes to highlight the importance
of the complementarity of neutrons with other experimental techniques and with
computational methods, and above all stimulate cross talk between research ﬁelds
and collaboration between scientists of different backgrounds. This is of increasing
importance with the trend in the ﬁeld moving to the study of more and more complex
systems, and with more difﬁcult interpretation of neutron data.
The book starts off by laying out the ground. First, the experimental techniques
available to probe the wide range of dynamics in Soft Matter systems are presented.
Existing computational methods are then reviewed, ranging from ﬁrst-principle
calculations to mesoscopic simulations. The core of the book is organized in three
sections, mainly in increasing order of system complexity, but to some extent there
is also a correlation with the evolution of neutron techniques.
The ﬁrst section deals with the dynamicsin traditional macromolecules, i.e.,
in polymer systems. Gabrys and Kanaya (Chap. 3) introduce the vast range of
motions possible and explain how neutrons have helped to distinguish them. Arbe
and Colmenero (Chap. 4) move on to the unusual dynamical properties observed
in polymer blends and relate these to the more fundamental phenomena of the
glass transition in amorphous liquids. Chapter 5 treats the speciﬁc example of
understanding the dynamics in solid polymer electrolytes as part of the move
to greener and lighter batteries. This section ﬁnishes off with the dynamics at
longer lengthscales, and provides a transition from polymers to the biological
macromolecular world of proteins and lipids covered in the second section.
Three of the chapters in this section deal with the active research ﬁeld of the
structure-dynamics-function relationship of proteins. Smith (Chap. 7) considers
neutron data in combination with simulations, Longeville and Doster (Chap. 8)
discuss the dynamic processes occurring in proteins, and Wood and Weik (Chap. 9)
emphasize the role of hydration water for protein function. The section ﬁnishes with
a chapter devoted to the dynamics of the lipid membranes which form an integral
part of cells and living organisms.
In the last section of the book we add extra parameters to the discussion,
namely we highlight the importance of conﬁnement, both soft and hard, and surface
effects. Chapter 11 presents the emerging technique of time-resolved small angle
scattering for studying kinetics. The effects of conﬁnement in Soft Matter systems
are discussed in detail with three examples: soft conﬁnement using micellar systems
(Chap. 12), nanoparticles in polymer matrices (Chap. 13), and “harder” conﬁnement
in nano/mesoporous materials (Chap. 14). The book ﬁnishes with examples in
the ﬁeld of shear dynamics in liquids and discusses the potential of an emerging
technique, grazing incidence small angle neutron scattering (Chap. 15).

Preface vii
The importance of neutrons in Soft Matter research is well understood within
the scientiﬁc community and this is shown by the development directions of new
neutron facilities. Not only are powerful new sources optimized for this type of
research, with corresponding new instruments being built, but also the complemen-
tary tools such as the development of deuteration and computation laboratories are
being set-up. There is huge scope for the new generation of scientists in the ﬁeld of
Soft Matter and dynamics, and neutrons will continue to play an important role in
answering the many questions that will arise.
Didcot, Oxon, UK Victoria Garc ´ıa Sakai
Orsay, CX, France Christiane Alba-Simionesco
Cambridge, MA, USA Sow-Hsin Chen

Contents
1 Experimental Techniques for Studies of Dynamics in Soft Materials1
Alexei P. Sokolov and Victoria Garc´ıa Sakai
2 Computational Tools to Understand Inelastic and
Quasielastic Neutron Scattering Data.................................... 25
Mark R. Johnson, Miguel A. Gonz´alez, Mohamed Zbiri,
and Eric Pellegrini
Part I Macromolecules: Polymers
3 Basic Modes of Motion in Polymers...................................... 59
Barbara J. Gabrys and Toshiji Kanaya
4 Complex polymers.......................................................... 103
Arantxa Arbe and Juan Colmenero
5 Solid Polymer Electrolytes................................................. 123
Janna K. Maranas
6 Future Perspectives: Moving to Longer Length and Time
Scales, from Polymers to Biological Macromolecules.................. 145
Dieter Richter
Part II Bio-Macromolecules: Proteins and Lipids
7 Structure and Dynamics of Biological Systems: Integration
of Neutron Scattering with Computer Simulation...................... 189
Jeremy C. Smith, Marimuthu Krishnan, Loukas Petridis,
and Nikolai Smolin
8 Protein Dynamics and Function.......................................... 205
St´ephane Longeville and Wolfgang Doster
ix

x Contents
9 Bio-Macromolecules and Hydration Water Dynamics................. 247
Kathleen Wood and Martin Weik
10 Lipid Membrane Dynamics............................................... 263
Maikel C. Rheinst¨adter
Part III Extra Complexity: Surface Effects and Conﬁnement
11 Application of Time-Resolved Small Angle Neutron
Scattering to Non-Equilibrium Kinetic Studies......................... 289
Jitendra P. Mata, William A. Hamilton, and Elliot P. Gilbert
12 Understanding the Stability of Micellar Systems of
Interest for the Study of Glasses, Freezing and Soft Conﬁnement.... 319
Tinka Spehr and Bernhard Frick
13 Structure and Dynamics of Polymer Nanocomposites
Involving Chain-GraftedSpherical Nanoparticles...................... 349
Peter F. Green, Hyunjoon Oh, Pinar Akcora, and Sanat
K. Kumar
14 Surface and Conﬁnement Effects in Nano/Mesoporous Materials.... 367
Jean-Marc Zanottiand Denis Morineau
15 Shear Dynamics: Understanding Boundary Slip
and Anomalies in the Structural and Dynamical
Properties of Liquids Under Flow........................................ 411
Max Wolff
Index............................................................................... 439

Contributors
Pinar AkcoraDepartment of Chemical Engineering, Columbia University,
New York, NY, USA
Arantxa ArbeCentro de F´ısica de Materiales (CSIC-UPV/EHU) – Materials
Physics Center (MPC), Paseo Manuel de Lardizabal 5, 20018 San Sebasti´an, Spain
Juan ColmeneroCentro de F´ısica de Materiales (CSIC-UPV/EHU) – Materials
Physics Center (MPC), Paseo Manuel de Lardizabal 5, 20018 San Sebasti´an, Spain
and Donostia International Physics Center, Paseo Manuel de Lardizabal 3, 20018
San Sebasti´an, Spain
Wolfgang DosterTechnische Universit¨at M¨unchen, Physik Department E 13,
James Franck Strasse 1, D-85747 Garching, Germany
Bernhard FrickInstitut Laue-Langevin, 6, rue Jules Horowitz, F-38042 Grenoble,
France
Barbara J. GabrysDepartment of Materials, University of Oxford, Oxford, UK
Elliot P. GilbertBragg Institute, Australian Nuclear Science and Technology
Organisation, Menai, NSW, Australia
Miguel A. Gonz´alezInstitut Laue Langevin, 6, rue Jules Horowitz, F-38042
Grenoble, France
Peter F. GreenDepartment of Materials Science and Engineering, University
of Michigan, Ann Arbor, MI, USA
William A. HamiltonBragg Institute, Australian Nuclear Science and Technology
Organisation, Kirrawee DC, NSW, Australia
Mark R. JohnsonInstitut Laue Langevin, 6, rue Jules Horowitz, F-38042
Grenoble, France
Toshiji KanayaDivision of Multidisciplinary Chemistry, Polymer Materials
Science, Kyoto University, Uji, Kyoto-fu, Japan
xi

xii Contributors
Marimuthu KrishnanOak Ridge National Laboratory, Oak Ridge, TN, USA
Sanat K. KumarDepartment of Chemical Engineering, Columbia University,
New York, NY, USA
St´ephane LongevilleLaboratoire L´eon Brillouin, CEA Saclay, F-91191 Gif sur
Yvette Cedex, France
Janna K. MaranasDepartment of Chemical Engineering, The Pennsylvania State
University, University Park, PA, USA
Jitendra P. MataBragg Institute, Australian Nuclear Science and Technology
Organisation, Kirrawee DC, NSW, Australia
Denis MorineauInstitute of Physics of Rennes, CNRS-University of Rennes 1,
Rennes, France
Hyunjoon OhDepartment of Materials Science and Engineering, University
of Michigan, Ann Arbor, MI, USA
Eric PellegriniInstitut Laue Langevin, 6, rue Jules Horowitz, F-38042 Grenoble,
France
Loukas PetridisOak Ridge National Laboratory, Oak Ridge, TN, USA
Maikel C. Rheinst¨adterDepartment of Physics and Astronomy, McMaster Uni-
versity, 1280 Main Street, West Hamilton, ON L8S 4M1, Canada and Chalk River
Laboratories, Canadian Neutron Beam Centre, Chalk River, ON, Canada
Dieter RichterInstitut f¨ur Festk¨orperforschung, Forschungszentrum J¨ulich, D-
52425 J¨ulich, Germany
Victoria Garc´ıa SakaiISIS Facility, Rutherford Appleton Laboratory, Harwell
Science and Innovation Campus, Didcot, UK
Jeremy C. SmithOak Ridge National Laboratory, Oak Ridge, TN, USA
Nikolai SmolinOak Ridge National Laboratory, Oak Ridge, TN, USA
Alexei P. SokolovChemical Sciences Division, Oak Ridge National Laboratory,
Oak Ridge, TN, USA and Department of Chemistry, University of Tennessee
Knoxville, TN, USA
Department of Chemistry, University of Tennessee, Knoxville, TN, USA
Tinka SpehrInstitut f¨ur Festk¨orperphysik, TU Darmstadt, Hochschulstr.
8, D-64289 Darmstadt, Germany
Martin WeikInstitut de Biologie Structurale, Grenoble, France
Kathleen WoodDepartment of Biophysical Chemistry, University of Groningen,
Groningen, The Netherlands Bragg Institute, Australian Nuclear Science and
Technology Organisation, Menai, NSW, Australia

Contributors xiii
M. WolffDivision for Materials Physics, Department of Physics and Astronomy
Science, Uppsala University, 751 05 Uppsala, Sweden,
Jean-Marc ZanottiLaboratoire L´eon Brillouin, CEA-CNRS, Saclay, France
Mohamed ZbiriInstitut Laue Langevin, 6, rue Jules Horowitz, F-38042 Grenoble,
France

Chapter 1
Experimental Techniques for Studies
of Dynamics in Soft Materials
Alexei P. Sokolov and Victoria Garc´ıa Sakai
1.1 Dynamics in Soft Materials
The ﬁeld of Soft Materials is experiencing an explosive growth during recent
years due to a variety of current applications (including advanced plastics and
elastomers, liquid crystals) and the essentially unlimited potential of future applica-
tions (various kinds of “smart” materials, bio-materials, etc.). The deﬁnition ofSoft
Matterincludes broad classes of materials ranging from polymers, liquid crystals
and colloidal systems, to biological systems. There are particular properties that
differentiate Soft Materials from others:
1. The existence of a great variety of meta-stable states with comparable potential
energies and separated by relatively small energy barriers (comparable to a
fewkT)
2. As a result, there is always a delicate balance between the Entropic and Enthalpic
contributions to the free energy, both contributions playing an important role in
the properties of soft materials (in strong contrast with hard materials)
3. Strong thermal ﬂuctuations and high sensitivity of their structure to small
external ﬁelds and perturbations
4. Macroscopic softness of the materials (the reason for the name) that reﬂects
signiﬁcant structural rearrangements under relatively small mechanical forces
A.P. Sokolov ()
Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN USA
and Department of Chemistry, University of Tennessee Knoxville, TN, USA
e-mail:[email protected]
V. G a r c´ıa Sakai
ISIS Facility, Rutherford Appleton Laboratory, Harwell Science, and Innovation Campus,
Didcot, OX11 0QX, United Kingdom
e-mail:[email protected]
V. G a r c´ıaSakaietal.(eds.),Dynamics of Soft Matter: Neutron Applications,Neutron
Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-01,
© Springer Science+Business Media, LLC 2012
1

2 A.P. Sokolov and V. Garc ´ıa Sakai
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
–12
–10
–8
–6
–4
–2
0
2
Fast Dynamics
Methyls
Secondary
Segmental
Chain
log(τ)
1000/T [K
–1
]
Fig. 1.1Relaxation map of polyisoprene (PIP) includes (1) chain and (2) segmental relaxation
processes with strongly non-Arrhenius temperature variations; (3) a secondary relaxation; (4)
methyl group rotations with Arrhenius temperature dependence; and (5) a fast relaxation with a
rather weak temperature dependence of the characteristic relaxation time. Most of the data are
from [6,7]
The properties listed above emphasize that transitions between multiple metastable
states under relatively small perturbations or due to equilibrium ﬂuctuations are
the distinct characteristics of soft materials. So, thedynamics, i.e., the motions of
molecular (or other structural) units, are the key to the main macroscopic properties
of Soft Matter. The dynamical processes that take place in soft materials are
very complex and the microscopic mechanisms of many of these remain poorly
understood. Cooperativity and dynamic heterogeneities are characteristic features
of dynamics in soft materials [1–5]. However, even these concepts are still not well
deﬁned.
Typically, the dynamics of soft materials include multiple relaxation processes
on local and global scales. They span many (can be more than 15) decades in time
(frequency) and are highly sensitive to temperature and external pressure. Figure1.1
shows an illustrative example of the characteristic relaxation times found in the
classical polymer polyisoprene (PIP), the major component of natural rubber. There
are at least ﬁve relaxation processes that involve molecular motions at different
length scales, from global chain relaxations to very local methyl group rotations.
All of these relaxation processes reﬂect molecular motions in a very complex
potential energy landscape with a large distribution of energy minima and energy
barriers separating the energetic states. Studying the dynamics of soft materials thus
can be rather challenging.
The complexity of the dynamics in soft materials manifests itself not only
through the large number of relaxation phenomena, but also by the nature of their
behavior. Most of the relaxation processes in soft materials cannot be described by
a single exponential decay. They are usually strongly stretched and can be described

1 Experimental Techniques for Studies of Dynamics in Soft Materials 3
in the time domain by the Kohlrausch–Williams–Watts (KWW) relationship:
exp[−(t/
τ)
β
][8], whereτis the characteristic relaxation time andβis the stretching
parameter. In the frequency domain, the stretched processes are usually described
by Cole–Cole, Cole–Davidson, or Havriliak–Negami distribution functions [9].
Secondary relaxations in most cases have symmetrically stretched shapes that are
well described by the Cole–Cole distribution function, while the primary structural
relaxation usually has an asymmetric shape which is strongly stretched from the
high-frequency side [10]. In contrast, fast picosecond relaxation is stretched from
the low-frequency side [11]. The reasons for the stretched spectra can be (1)
dynamic heterogeneities that lead to a distribution of relaxation times, and/or (2)
intrinsically non-exponential relaxation processes. Most of the relaxation processes
also exhibit particular temperature dependences. Usually, only secondary relax-
ations follow an Arrhenius temperature dependence
τ=τ0exp(E/kT)(Fig.1.1).
However, chain and primary structural (segmental) dynamics exhibit much stronger
temperature variations that are traditionally described by the Vogel–Fulcher–
Tamman (VFT) equation:
τ=τ0exp[B/(T−T 0)]. The characteristic relaxation
times of the fast picosecond process do not change signiﬁcantly, down to very low
temperatures [11].
This chapter provides a brief overview of the experimental techniques commonly
used for analysis of the dynamics of soft materials. We realize that it is not possible
to cover all techniques in a single chapter and thus will focus on the traditional and
more broadly used techniques. We shall cover mechanical and dielectric relaxation
spectroscopy, Nuclear Magnetic Resonance (NMR), light and X-ray scattering
techniques. We will compare their advantages and disadvantages. The focus of this
book is to highlight the importance of neutron scattering techniques in probing the
dynamics in Soft Matter and so it will be put into context in this introductory chapter,
but we will not give a detailed description here. In fact, given the complexity of the
dynamics in soft materials, we wish to emphasize the complementarity between all
the techniques discussed here, and that it is crucial to combine information from sev-
eral of these techniques to fully understand the underlying mechanisms of molecular
motions. For more in-depth reading on the basics of neutron scattering, we refer the
reader to a detailed introduction on neutron scattering [12, and references there-in]
and on neutron instrumentation [13] and to the more speciﬁc applications in the
remit of Dynamics of Soft Matter that follow in other chapters of this book.
There are of course other techniques, which will not be discussed, for example,
time-resolved optical techniques (e.g., optical Kerr-effect [14]) that are used to
monitor relaxation processes. A variety of techniques are based on ﬂuorescence,
measuring the kinetics of ﬂuorescencedecay and change of its polarization.
Pump-probe techniques and forced Rayleigh scattering are actively used to study
molecular diffusion and other relaxation processes. Also, optical microscopy has
been actively used to study microscopic details of motions in colloidal systems. It
has helped to visualize cooperativity and dynamical heterogeneities in these model
soft materials [3].
Figure1.2presents a traditional view of the time (frequency) range accessible
to different techniques. It is important to note that scattering techniques have a

4 A.P. Sokolov and V. Garc ´ıa Sakai
10
3
10
2
10
1
10
0
10
–1
10
0
10
–2
10
–4
10
–6
10
–8
10
–10
10
–12
10
–14
10
–2
10
–1
10
0
10
1
10
2
10
1
10
3
10
5
10
7
10
9
10
11
10
13
X-PCS
Mechanical
Frequency [Hz]
Q [nm
–1
]
Dielectric spectroscopy
Neutron
Scattering
IXS
Light Scattering
Time [s]
Length [nm]
Fig. 1.2Sketch showing the frequency (time) and wavevector (length) ranges accessible with tra-
ditional experimental techniques. Longer times (not shown) can be accessed by many techniques.
IXS is Inelastic X-Ray Scattering and XPCS is X-ray Photon Correlation Spectroscopy. Dielectric
and Mechanical relaxation spectroscopy do not have a particular length scale(Q), thus are placed
outside of the main plot
signiﬁcant advantage due to an additional variable – the scattering angle. It provides
information on the geometry of the motion through the measurements of the
scattering wavevectorQ. Thus, in Fig.1.2, the lengthscale(∼2
π/Q)accessible to
experimental techniques is shown as thex-axis. Mechanical and dielectric relaxation
spectroscopies have no particularQ, and that is the reason they are outside of the
main sketch presented in Fig.1.2.
1.2 Neutron Scattering Spectroscopy
The dynamics of molecules at a molecular level can be probed directly byneutron
scattering spectroscopy. Like other scattering techniques, neutron spectroscopy
simultaneously probes the timescale of the motion as well as the lengthscale over
which the motion takes place. This enables us to obtain geometrical information and
provides a deeper insight into the nature of the dynamical processes that lead to the
viscoelastic and mechanical properties of Soft Matter systems. In this same context,
neutron scattering spectroscopy is a powerful technique owing to the overlap of
the time-length window it can access with that of computer simulations (molecular
dynamics and lattice methods). Thus, thereis a reciprocal advantage in combining
data from these two methods: experimental neutron scattering data can be used to

1 Experimental Techniques for Studies of Dynamics in Soft Materials 5
validate theoretical models which are becoming increasingly elaborate and need
to be more realistic given the increasing complexity of samples; and accurate
theoretical models can be used to interpret dynamical data from neutron scattering
experiments and create visualizations of the dynamical processes (see e.g., Chap. 2).
The motions probed by neutrons cover a very broad range, from the measurement
of momentum distributions on the fs time scale (eV), to slow dynamics such as the
relaxation of whole polymer chains in the melt in the∼100ns (neV) timescale. The
accessible scattering wavevector range allows analysis of geometry of molecular
motions on length scales from below an Angstrom to dozens of nanometers.
Traditional applications of neutron scattering spectroscopy to soft materials have
mainly focused on inelastic (energy transfer peaks centers at ﬁnite energies – to
study vibrational modes and complementing Raman or Infra-red spectroscopy) and
quasi-elastic scattering, at energies close to the elastic line (small energy transfers
centered at zero energy – to study rotational and diffusional processes which
complement dielectric relaxation spectroscopy and NMR) [15]. In addition, the
last 20 years have seen a signiﬁcant increase in the number of experiments using
Neutron Spin Echo (NSE), a technique which probes the longer time and length
scales that are relevant for many soft materials [16]. Dynamical information from
NSE nicely complements the structural data obtained from the very commonly used
technique of small angle-neutron scattering. Furthermore, NSE has provided new
insights into colloidal systems (Chap. 12) and offers new possibilities in surface
science (Chap. 14).
Neutrons have wavelengths and energies that are comparable with interatomic
spacings and molecular motions, and thus allow us to probe motions at a molecular
scale. The scattering cross-section is simple (in this energy range, it is just a
constant) and can be measured on an absolute scale, allowing direct and quanti-
tative comparison of neutron measurements with theory and modeling/simulation.
In addition and in comparison to X-rays, neutrons are highly penetrating and
nondestructive. Neutrons have no charge and a negligible dipole moment and so
can travel large distances before being scattered or absorbed. This allows studies
under external ﬁelds such as temperature, pressure, shear stress, etc. Due to their
low energy (usually in meV range), neutrons are also nondestructive and samples
can easily be irradiated for a long time (days) and reused after a neutron experiment.
This is of particular importance for biological materials. In particular for Soft Matter
studies, the large contrast achieved by the isotopic substitution of hydrogen (one
of the main components of soft materials) with deuterium, without changing the
intrinsic properties of the material, allows the selective study of speciﬁc units of
a molecule (e.g., peptide in a protein)or components in a multicomponent system
(e.g., polymer in solution). See for example Chap. 6, Section 2.1, for a “classical”
example of using isotopic substitution (also known as contrast variation) to study
single chain dynamics in liquid poly(ethyl ethylene).
There are of course some disadvantages to the technique. First of all, it is not
available to have on the bench of a university laboratory. Peer-reviewed beam-time
proposals have to be approved to obtain neutron-measuring time at neutron sources.

6 A.P. Sokolov and V. Garc ´ıa Sakai
Despite continuing improvements at neutron facilities, in neutron instrumentation
and optics, the brilliance of neutron beams is far lower than that from synchrotron
X-rays. As a result, experiments require relatively large sample sizes from hundreds
of mg’s to a few g’s which for many biological samples is very hard to achieve.
All in all, neutron scattering spectroscopy has unique advantages over other
spectroscopic techniques. However, its low accuracy (statistics), limited frequency
(time) window, and limited access to spectrometers require extensive use of com-
plementary techniques to get a more detailed and accurate picture of the underlying
dynamics.
In what follows, we present the measurable quantities of a neutron experiment in
the context of dynamics. When a neutron is scattered by a nucleus, it can change its
energy as well as its momentum such that:
ΔE=E
f−Ei=±¯hω=
¯h
2m
τ
− →
k
2
f
−
−→
k
2
i
β
, (1.1)
¯h
τ
− →
k
f−
−→
ki
β
=¯h
− →
Q, (1.2)
wherek=2
π/λis the neutron wavevector (momentum) andλis the neutron
wavelength,mis the neutron mass,Eis the energy,Qis the momentum transfer
(Q=k
f−ki), and i and f refer to the initial and ﬁnal states of the scattering process.
In dynamical experiments, we measure the double differential cross-section, i.e., the
intensity of scattered neutrons with an energy changeΔEinto a solid angleΩ:
∂
2
σ
∂Ω∂E
∝
k
f
ki
N[σcohScoh(Q,ω)+σincSinc(Q,ω)], (1.3)
whereNis the number of scatterers,
σcohandσincare called the coherent and inco-
herent cross-sections andS
coh(Q,ω)andS inc(Q,ω)are the coherent and incoherent
scattering laws, respectively (also sometimes known as dynamic structure factors).
The coherent and incoherent cross-sections are calculated from a scattering length
speciﬁc to each isotope of a particular chemical element. The dynamic structure
factors are related to the intermediate scattering functions,I
coll(Q,t)andI self(Q,t),
which are the time Fourier transform of the “distinct” and “self” parts of the Van
Hove correlation function:
S
coh(Q,ω)=
1
2π
π
+∞
−∞
Icoll(Q,t)exp(−i ωt)dt, (1.4)
S
inc(Q,ω)=
1
2π
π
+∞
−∞
Iself(Q,t)exp(−i ωt)dt, (1.5)
I
coll(Q,t)=
1
N
∑
k
∑
l
πexp{−iQr k(0)}exp{−iQr l(t)}Δ, (1.6)
I
self(Q,t)=
1
N
∑
k
πexp{−iQr k(0)}exp{iQr k(t)}Δ. (1.7)

1 Experimental Techniques for Studies of Dynamics in Soft Materials 7
In the classical limit, the “distinct” part is the probability of ﬁnding a particleiat
timetat a distancerfrom a position of a particlejat timet=0, and the “self” part is
the probability of ﬁnding a particleiat timetat a distancerfrom its position at time
t=0. Thus, incoherent scattering in a dynamical measurement gives information
on self-dynamics (the dynamics of individual atoms) and coherent scattering gives
information on collective motions.
From hydrogenated samples, we mainly measure incoherent scattering, owing
to the incoherent cross-section of hydrogen being 40 times larger than the co-
herent cross-section. The appropriate technique is quasi-elastic neutron scattering
(QENS) which probes very small changes in the neutron energy, in the ueV range.
Quasi-elastic neutron spectrometers cover timescales between 0.1 ps and 4 ns and
lengthscales between 1 and 30˚A, probing motions such as vibrations (including the
Boson peak), rotations of small molecular units, localized motions, and structural
relaxations. The dynamic structure factor can be analyzed in the frequency domain,
or Fourier transformed to the time domain and modeled with an appropriate
dynamical model. Not only temporal information such as diffusion coefﬁcients and
characteristic relaxation times can be extracted, but the so-called Elastic Incoherent
Structure Factor (EISF) can also be calculated and ﬁtted to models to obtain the
geometry of the motion. The EISF is deﬁned as the ratio of elastic scattering to total
scattering. Finally, for the study of collective dynamics at the nanosecond timescale,
the most efﬁcient technique is NSE whichmeasures the intermediate scattering
function in the time domain directly.
From deuterated samples, we mainly measure coherent scattering, which is more
difﬁcult to interpret since it involves the correlated motions of many atoms. Indeed,
for systems as complex as most soft matter, a detailed interpretation requires the
aid of some form of atomistic simulation. However, by selective hydrogenation
of speciﬁc parts of an otherwise deuterated system, we can then selectively pick
out the self-dynamics of those atoms. If the number of H atoms is sufﬁcient then
their incoherent scattering dominates the signal and this can be done directly. If the
number of H atoms is small, then it may be necessary to measure the difference
between the scattering from fully and partially deuterated systems.
1.3 Mechanical Relaxation Spectroscopy
The mechanical properties of soft materials are one of the most important pa-
rameters for many practical applications. Knowledge of the mechanical properties
and their time and temperature dependencies are crucial for these applications.
Traditionalmechanical relaxation spectroscopymeasures stress under applied
strain, or the reverse, strain under applied stress. The measurements may be
in the time domain or in the frequency domain (dynamic mechanical measure-
ments). Time domain measurements usually cover relaxation times from days
to milliseconds. Traditional dynamic mechanical measurements cover frequencies

8 A.P. Sokolov and V. Garc ´ıa Sakai
10
–4
10
–3
10
–2
10
–1
10
0
10
1
10
2
10
3
10
4
10
5
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
Chain modes
Segmental
relaxation
G'(ω) and G''(ω) [Pa]
ωa
T
G'
G''
Fig. 1.3Frequency dependence of the storage and loss shear moduli, G
ω
and G
ωω
, respectively,
for polystyrene of molecular weight 10,000 g mol
−1
. The spectra are constructed by combining
measurements at different temperatures using the shift factora
Tand assuming time-temperature
superposition (TTS). Data are taken from [18]
between∼10
−6
Hz and∼10
3
–10
4
Hz. However, there are also resonance and wave
propagating techniques that can measure mechanical properties in the MHz–GHz
frequency range. Brillouin light scattering probes mechanical properties in the
GHz frequency range. In addition, achievements in inelastic X-ray scattering
spectroscopy (IXS) have opened up the possibility of mechanical measurements
even in the THz range [17] (though mostly for longitudinal waves). Thus, combining
different techniques one can almost entirely cover the time range from picoseconds
to many hours, albeit not necessarily continuously.
Figure1.3shows mechanical storage and loss moduli for the polymer
polystyrene. The modulus has a complicated spectrum with contributions from
segmental dynamics and chain (Rouse) modes. The shear modulus of the segmental
dynamics has an amplitude in the GPa range, while the modulus of chain modes
is signiﬁcantly lower in the MPa range. This difference is due to the variation
in energetic (segmental) and entropic (chain) driving forces and is typical for
polymers. Figure1.3clearly demonstrates that polymer dynamics span a very large
frequency range. Unfortunately, it is notpossible to cover such a broad frequency
range continuously with any current mechanical relaxation technique. As a result,
researchers measure the spectra in the accessible frequency range (usually covering
3 to 5 orders in time) at different temperatures and then construct a master curve
similar to that shown in Fig.1.3, assuming the time-temperature superposition
(TTS) principle holds [19]. It is well-known, however, that TTS breaks down for
most polymers [20–22]. Master curves should therefore be considered as qualitative
and one must be cautious when performing a quantitative analysis.

1 Experimental Techniques for Studies of Dynamics in Soft Materials 9
The main disadvantage of mechanical relaxation spectroscopy is the absence of
any microscopic information on the molecular motion underlying the relaxation
process. The measured quantities (usually modulus and strain) have no direct
relationship to molecular motions and many model assumptions are usually involved
in the data interpretation. However, details of the real molecular motions are
missing. Mechanical relaxation can measure bulk and shear moduli which provide
information on mechanical relaxation under compression or deformation. This
difference gives some additional hints as to the underlying mechanisms of the
relaxation, but no microscopic details. Having said that, one can design some
experiments where, by careful modiﬁcation of the molecular structure, one can
gain some microscopic insight even from mechanical measurements. An illustrative
example of this is the work by Yee et al. on the nature of the secondary relaxation
(
γ-process) in polycarbonates [23,24]. By designing tailored block co-polymers,
the authors were able to demonstrate the existence of signiﬁcant cooperativity along
the chain involved in a single
γ-relaxation event [23,24]. This approach, however,
requires signiﬁcant effort and precise chemistry of materials.
The advent of atomic-force microscopy (AFM) brings new possibilities to
studies of the mechanical properties of materials. It provides information on
the mechanical moduli, energy dissipation, and viscoelasticity of surfaces at the
nanoscale [25]. Recent works involving pulling single molecules have also created
signiﬁcant interest in single moleculemechanical relaxation spectroscopy [26,27].
This technique has already been applied to studies of proteins and other biological
systems [26,27]. Still, the use of AFM techniques in the analysis of dynamics in
soft matter (even on surfaces) remains limited and is not well developed. However,
there could be signiﬁcant progress in this direction in the near future.
1.4 Dielectric Relaxation Spectroscopy
One of the most powerful relaxation spectroscopy techniques isdielectric spec-
troscopy. This covers a very wide frequency range. Many laboratories now have
dielectric spectroscopy instrumentation that can probe motions ranging from molec-
ular vibrations (infra-red spectroscopy) to diffusion on the timescale of hours
(micro-Hertz) [28]. Thanks to the great progress in electronics and radio-physics,
dielectric spectroscopy provides measurements with very high accuracy covering
a broad frequency range in a relatively short time. However, like mechanical
relaxation measurements, its main disadvantage is the absence of microscopic
information about the molecular motions. Dielectric spectroscopy measures the
dielectric constant (time or frequency dependence), but no spatial scale information.
In addition, the electrical conductivity can mask some of the signal. In the following,
we present a general overview. Further details can be found in general textbooks on
the subject [29,30].
In simple terms, dielectric relaxation spectroscopy measures reorientation of
dipoles and translational motion of charges. It is usually analyzed in terms of the

10 A.P. Sokolov and V. Garc ´ıa Sakai
dielectric constant which in the frequency domain will have a real and an imaginary
part (similar to the mechanical storage and loss modulus):
ε(ω)=ε
ω
(ω)−iε
ωω
(ω)+i
σω
ε
vac. (1.8)
Here
ε(ω)is the complex permittivity,σis the electrical conductivity, andεvac
is the dielectric constant of vacuum. Application of an electric ﬁeld leads to the
reorientation of dipoles associated with molecular units. The complex permittivity
can also be expressed as:
ε(ω)−ε∞
ε0−ε∞
=1−i ω
1
π
π
∞
0
Φ(t)exp(−i ωt)dt, (1.9)
where
ε0andε∞are the limiting values of the dielectric permittivity at 0 and
inﬁnite frequencies, respectively, andΦ(t)is a macroscopic relaxation function that
for a single dipole
μcan be expressed as the time autocorrelation function of its
reorientation:
Φ
μ(t)=πμ(0)μ(t)Δ/μ
2
=πcosθ(t)Δ. (1.10)
θ(t)is the angle between the dipole vector att=0andt=t. In reality, systems
have local ﬁeld corrections and many dipoles; in this case, the contribution from all
dipoles is measured asM(t)=∑
iμi(t).Then
Φ
μ(t)=
πM(0)M(t)Δ
πM(0)M(0)Δ
=
∑
i
∑
j
πPi(0)Pj(t)Δ
∑
i
∑
j
πPi(0)Pj(0)Δ
, (1.11)
whereP
i(t)is the instantaneous dipole moment of moleculei.Asusual,the
correlation function can be divided into “self”(i=j)and “distinct”(iλ =j).The
major contribution is that from the “self” correlation term, and the cross-correlations
(“distinct”) are neglected in many cases.
Certain structures (e.g., some polymers or
α-helixes in proteins) exhibit a
persistent cumulative dipole moment along the structure. In this case, there may be
a number of separate dielectric responses: (1) individual responses from structural
units (e.g., monomers, residues) and (2) a collective response that is accumulated
along the structure. Some polymers (e.g., polypropylene oxide, polyiosprene,
polyoxybutylene) have the dipole moment accumulating along the chain contour
together with a transverse dipole moment component of each monomer. Such
polymers exhibit at least two dielectric relaxation processes, though there can
be even more due to secondary relaxations: (1) a high-frequency process due to
segmental motions of the transverse component
μ
⊥
of the dipole moments of the
monomers and (2) a low-frequency process due to the long-range motions of the
cumulative dipole moment along the chain. In this case, the correlation function for
the entire chain will be:

1 Experimental Techniques for Studies of Dynamics in Soft Materials 11
10
–3
10
–1
10
1
10
3
10
5
10
7
10
–2
10
–1
Segmental
Chain
Secondary relaxation
206K
290K
225K
ε"(ω)
frequency [Hz]Fig. 1.4Dielectric relaxation spectra of polyisoprene (PIP) with a molecular weight of
10 kg mol
−1
at a few selected temperatures. The two strong peaks correspond to the chain (lower
frequency) and segmental (higher frequency) dynamics [6]. The high-frequency shoulder of the
spectrum atT=206 K corresponds to the secondary relaxation. We emphasize that no TTS is
required to measure dielectric relaxation spectra in such a broad frequency range
Φμ(t)=
μ
2
πRi(0)Ri(t)Δ+∑
i
∑
j
Δ
μ
⊥
i
(0)μ
⊥
j
(t)
ω
μ
2
πRi(0)Ri(0)Δ+∑
i
∑
j
Δ
μ
⊥
i
(0)μ
⊥
j
(0)
ω, (1.12)
where
μis the parallel component of the monomer’s dipole moment per unit
length and<R
i(0)Ri(t)>is the correlation function of the end-to-end vector. Thus,
dielectric spectroscopy can be used to measure the relaxation of the end-to-end
vector, which is one of the main parameters in theories of polymer chain dynamics.
However, as has been emphasized above, this is only possible for polymers that have
dipole moments accumulating along the chains.
Similar effects will appear in the dielectric relaxation spectra of proteins and
polypeptides. The
α-helical structure creates a cumulative dipole moment along its
axis. As a result, the total dielectric response will have contributions from individual
residues as well as from the whole
α-helix [31].
Figure1.4shows the dielectric relaxation spectra of polyisoprene (PIP). The low-
est frequency peak corresponds to the chainrelaxation, while the higher-frequency
peak is that due to the segmental relaxation. Comparing Figs.1.3and1.4, it is clear
that the relative amplitudes of the chain and segmental dynamics to the relaxation
spectrum differ signiﬁcantly between mechanical and dielectric spectroscopy. If in

12 A.P. Sokolov and V. Garc ´ıa Sakai
mechanical relaxation these contributions are deﬁned by the ratio of entropy-driven
(chain) to energy-driven (segmental) forces and will be always low(∼10
−3
),the
dielectric response depends on the amplitudes of the dipole moments (1.12). As a
result, the ratio of the amplitudes of chain to segmental peaks in dielectric spectra
varies signiﬁcantly between polymers and the amplitude of the chain mode can be
even higher than the segmental one (as for PIP in Fig.1.4).
The high-frequency shoulder of the segmental peak (Fig.1.4) corresponds to the
secondary relaxation which is dielectrically active in PIP. We want to emphasize
that only molecular motions that affect dipole moments can be measured with
this technique. For example, PIP has active methyl group dynamics (as seen
in Fig. 2.1). However, since they have no dipole moment, it does not appear
in the dielectric relaxation spectra. Another interesting example is poly(methyl
methacrylate) (PMMA), which has a side group with a strong dipole moment.
As a result, the amplitude of a secondary relaxation in PMMA is signiﬁcantly
higher than that of the segmental dynamics and thus dominates the dielectric
spectra [32].
To conclude, we emphasize the power of dielectric spectroscopy in provid-
ing accurate measurements of relaxation spectra over a broad frequency range.
However, the absence of any microscopic information signiﬁcantly complicates
interpretation. An example of this is for measurements of protein solutions. At least
three additional peaks (compared to bulk water) appear in the dielectric relaxation
spectra. The slowest of these is ascribed to protein rotation (tumbling). However, the
assignment of the higher-frequency modes remains controversial [33–38]. In early
works, both processes were assigned to relaxations of water molecules [33,34]:
the fastest process was ascribed to the dynamics of hydration water which is only
∼3–5 times slower than that in bulk water, while the slower process was assigned
to tightly bound hydration water moving∼100 times slower than bulk water. The
assignment of the slower process, however, was questioned in later works [35–
39]. It was then ascribed to protein motions [36–38] and to a protein–hydration
water cross-term [35]. In fact, even the assignment of the fast process was recently
questioned [39]. Based on comparisons between neutron and dielectric spectroscopy
data, the authors argued that this process is due to coupled protein–hydration water
relaxation. This example clearly illustrates the difﬁculties in interpreting dielectric
spectra: not only is a microscopic interpretation not feasible, but also the assignment
of relaxation modes can be questionable. Neutron scattering measurements can
alleviate some of these difﬁculties by providing a detailed microscopic picture
to complement the relaxation times and spectral shapes obtained from dielectric
spectroscopy, which can in turn be fed back into the analysis of neutron scattering
data [39].

1 Experimental Techniques for Studies of Dynamics in Soft Materials 13
1.5 NMR Relaxation Spectroscopy
NMR spectroscopyis a very powerful and well-developed technique for studying
the structure and dynamics in a number of materials. It measures the motion of
nuclear spins (i.e., almost directly atomic motions). There are many good textbooks
describing the basics of NMR spectroscopy and here we will only present a brief
overview of the main principles in the context of soft matter.
Each nucleus has a spinI. Application of a magnetic ﬁeldBleads to a split of
the energy levelsΔE=
μB/Iwhereμis the magnetic momentum of the nucleus.
Transitions between levels can be stimulated and monitored using resonance radio-
frequency. Different nuclei have differentresonance frequencies. There is proton
NMR (for H-atoms),
13
C NMR, etc. Different isotopes of the same atom have
different resonance frequencies and this effect is often used for selective studies
of particular parts of the molecule. For example, regular isotopes
12
Cand
16
Ohave
I=0 and do not contribute to the NMR signal, whereas
13
Cand
17
O do, and thus
can be used to selectively study a desired part of the molecule.
For the same atom, NMR resonance frequencies depend on the neighboring
atoms (chemical binding). For example, the resonance frequency of the H-atom is
different for CH
3,CH2, and OH groups. In addition, there is NMR ﬁne structure that
reﬂects differences in the nuclear resonance frequency due to spin–spin interactions
of neighboring nuclei of the same type. All these details allow very selective studies
of particular atoms of the molecule. Using NMR, one can study not only the average
motion of all atoms, for example hydrogen, but also of particular atoms, such as
H-atoms in methyl groups. Selectivity can also be achieved with neutron scattering,
where the difference in scattering interaction between the hydrogen and deuterium
atoms is exploited to highlight different parts of a molecule. Thus, information from
both techniques can be used in a complementary fashion to obtain a more complete
dynamical picture in soft materials.
Spin systems have an equilibrium population of energy levels in a magnetic
ﬁeld. This population can be disturbed by aresonant radio-frequency. The resulting
nonequilibrium spin system can relax back toits original equilibrium state through
the so-calledspin–lattice relaxation, characterized by a timeT
1which is the average
time taken for an individual nucleus to return to its equilibrium state. The “lattice”
in this case refers to the other molecular units surrounding the nucleus that provide
energy exchange through various molecular motions. The spin of each nucleus also
has a precession about the direction of the magnetic ﬁeld. This precession induces a
magnetic ﬁeld that might interact with the spins of other nuclei which are precessing
at the same frequency. This spin–spin interaction does not change the overall energy
of the spin system, but it shortens the lifetime of the spin states and broadens their
spectra. The spin–spin relaxation is characterized by a timeT
2.
Molecular motions produce ﬂuctuations of the local magnetic ﬁeld. In this way,
NMR relaxation can provide information about molecular motions on very local
(atomic) scale. This relaxation depends strongly on the nucleus. Using
13
C,
15
N
and H atoms provide a rather efﬁcient method to study dynamics. There are

14 A.P. Sokolov and V. Garc ´ıa Sakai
many different variations of the NMR spectroscopy technique that are used to
study the dynamics of soft materials. Let us ﬁrst consider the example of pulsed-
ﬁeld gradient NMR spectroscopy which is a tool for the study of molecular
diffusion. Traditionally, this uses a magnetic ﬁeld with the gradient along one
axis. As a result, the nuclear resonance frequency is a function of position along
the ﬁeld gradient. Analyzing the NMR frequency as a function of time after the
pulse provides a measure of molecular diffusion without using any markers. This
technique is effective for the analysis of molecular diffusion in low-to-moderate
viscosity materials [40]. The limitation is given by the time it takes to analyze the
spin frequency before it relaxes and the strength of the ﬁeld gradient.
Multidimensional NMR, based on multiple pulses with varying frequencies
and time sequence between pulses, has provided many interesting results for
soft materials [41–45]. In particular, multidimensional NMR has emphasized
the importance of dynamic heterogeneities in the structural relaxation of glass-
forming liquids [43,44]. Unlike most other spectroscopic techniques used for
dynamical studies, which measure two-point correlation functions and an average
relaxation of the measured variable, 3- and 4-dimensional NMR can measure
higher order correlation functions and can probe dynamic heterogeneities [43,44].
These studies reveal the existence of dynamic heterogeneities (sub-ensembles of
molecules relaxing with different relaxation times) in the structural relaxation at
temperatures close to the glass transition temperature [43,44]. The same studies
demonstrate that these heterogeneities are short lived and faster relaxing molecules
may become the slowest on time scales comparable to the average relaxation time
[44]. Multidimensional NMR has been also used to estimate the length scale of
the dynamic heterogeneities, which at temperatures close toT
gis between 1.5 and
3.5 nm [45].
Despite signiﬁcant progress in NMR spectroscopy and the availability of a wide
variety of NMR techniques, it suffers from the very strong localization of the probe –
the spin of the nucleus. As a result, the dynamics probed by NMR are very localized,
associated with the motion of speciﬁc atoms. This requires very strong model
assumptions, so very sophisticated NMR measurements are needed to analyze large-
scale motions and/or cooperative motions of many structural units. In addition, the
technique does not provide a broad andcontinuous frequency dynamical window
and is usually effective in relatively narrow spectral ranges. However we should
mention that very recent developments in ﬁeld-cycling NMR [46] might open
a possibility for a broadband NMR spectroscopy. Finally, interpretations of the
measuredT
1andT 2are often not straightforward and strongly model-dependent.
1.6 Light Scattering
Light scattering is another experimental technique that is broadly used to study
dynamics of soft materials. As with any scattering technique, it has the advantage of
measuring relaxations and vibrational spectra as a function of the scattering angle
θ.

1 Experimental Techniques for Studies of Dynamics in Soft Materials 15
The latter deﬁnes the scattering wavevectorQ≈2Q isin(θ/2),whereQ iis the
wavevector of the incident light wave (note that change of the light photon energy
during the scattering process is usually negligible). The analysis of light scattering
spectra as a function ofQprovides geometric details of molecular motion. The range
of accessibleQis rather narrow, between 10
−4
and 10
−2
nm
−1
, owing to the large
wavelength of light (between 300 and 800 nm in the visible range). As a result, it
does not allow studies of the geometrical details of motions at the molecular level
(as is the case with neutron and X-ray scattering), but is very efﬁcient for studies of
colloidal systems.
In most cases, high signal-to-noise ratio data are obtained with light scattering
over a broad frequency (time) window and in a relatively short time (though in
this context it still cannot in general compete with dielectric spectroscopy). An
additional advantage of light scattering is the very small probe size. One can
focus light to∼1μm and use it either for studies of extremely small samples (of
the order of picograms) or for the analysis of the signal across the sample with
resolution in space better than a micron. In most cases, the measured spectra
do not require any signiﬁcant corrections. Furthermore, light scattering has an
additional parameter, the polarization of the scattered light. Scattered light can be
polarized,I
, having the same polarization as the incident light, or depolarized,
I
⊥, with the polarization perpendicular to the incident light polarization. Analysis
of the depolarization ratioI
⊥/I
provides additional information about the type of
molecular motion taking place: for example, a number of rotational processes will
cause depolarized scattering while isotropic ﬂuctuations such as those from density
or chemical composition usually lead to polarized scattering.
A signiﬁcant disadvantage of the light scattering technique is that it measures
ﬂuctuations of the optical polarizability (in simple terms, ﬂuctuations of the
refractive index) caused by molecular motions, rather than the molecular motions
directly. As a result, signiﬁcant model assumptions are involved in the interpretation.
In addition, transparent samples are required in many cases and a high ﬂuorescence
signal (either intrinsic or caused by impurities) usually destroys the accuracy of light
scattering measurements.
There are three major light scattering techniques:
1. Raman spectroscopy which usually covers a wavenumber range between 3–5 and
∼5,000cm
−1
(frequency range from∼100GHz up to∼200THz) and is used
mostly for studies of vibrational modes and fast ps ﬂuctuations
2. Interferometry (Brillouin spectroscopy) that covers a frequency range from
∼100MHz up to∼1THz (∼30cm
−1
)and is used for studies of acoustic
vibrational modes (e.g., mechanical properties at GHz frequencies) and relax-
ation processes in the ps–ns time range
3. Photon Correlation Spectroscopy (PCS) (often called Dynamic Light Scattering)
that works in the time domain and covers motions from about 10 ns (optimistic
estimate) to hours. By combining these three techniques, one can essentially
cover the entire frequency (time) range with a small gap remaining in the
ns-regime [47]

16 A.P. Sokolov and V. Garc ´ıa Sakai
A detailed theoretical description of light scattering can be found in [48]. In the
following, we give only a short overview. In simple terms, light scatters from local
ﬂuctuations of the dielectric constant
δεyz(r,t)(indicesyzrefer to the two axes of the
average light polarization). Light scattering measures the intensity of the scattered
lightI(Q,
ω):
I(Q,
ω)∝Q
4
I0
π
exp(−i ωt)πδε(Q,t)δε(−Q,0)Δdt. (1.13)
HereI
0is an incident intensity andδε(Q,t)is the Fourier transform of the dielectric
constant ﬂuctuations in space. We have omitted the polarization indices in the
equation for simplicity. Due to extremely small variations of the value of the
scattering wavevectorQ,thetermQ
4
is usually neglected. The main challenge in
the interpretation of light scattering spectra is to ﬁnd the connection between the
measured
δε(Q,t)and the underlying molecular motion.
The main approaches used to describe ﬂuctuations in the dielectric constant can
be divided into two groups:
(a) Continuum approximation
This approach considers
δε(r,t)as a continuous function ofrwithout taking into
account molecular polarizability, based onthe long wavelength of light relative
to the characteristic molecular scales. In this case,
δεyz(r,t)for an isotropic one
component system can be expressed in terms of an elasto-optical coefﬁcientaand
an local deformation tensor
γyz(r,t)such that:
δεyz(r,t)=a 1γiso(r,t)+a 2˜γyz(r,t), (1.14)
where
γiso(r,t)is the isotropic part of the deformation tensor and˜ γyz(r,t)=
γyz(r,t)− δyzγiso(r,t)/3 represents the off-diagonal elements of the tensor. There
are contributions due to isotropic compression and due to shear deformations. For
isotropic ﬂuctuations, the scattering intensity as a function of time is given by:
I
iso(Q,t)∝π δεiso(Q,t)δεiso(−Q,0)Δ=a
2
1
πγiso(Q,t)γiso(−Q,0)Δ,
∝
λ
∂ε
∂ρ
Ω
2
πδρ(Q,t)δρ(−Q,0)Δ=
λ
∂ε
∂ρ
Ω
2
S(Q,t), (1.15)
where
δρare ﬂuctuations in densityρ. Thus, for isotropic ﬂuctuations, light
scattering measures the same intermediate scattering function as in, for example,
coherent neutron scattering, but multiplied by the elasto-optical coefﬁcient. As
usual, the frequency dependent intensity,I(Q,
ω), is the Fourier transform ofI(Q,t).
Now let us consider the example of isotropic ﬂuctuations in a two-component
system. In that case:
δε=
λ
∂ε
∂ρ
Ω
T,C
δρ+
λ
∂ε
∂C
Ω
T,ρ
δC+
λ
∂ε
∂T
Ω
ρ,C
δT, (1.16)

1 Experimental Techniques for Studies of Dynamics in Soft Materials 17
whereδCandδTcorrespond to ﬂuctuations in concentration and temperature,
respectively. The third term in the (1.16) is usually negligible. If the ﬂuctuations
of density and concentration are not correlated, then two contributions can be
separated as:
π
δε
2
Δ=
λ
∂ε
∂ρ
Ω
2
T,C
πδρ
2
Δ+
λ
∂ε
∂C
Ω
2
T,
ρ
πδC
2
Δ. (1.17)
Concentration ﬂuctuations are usually signiﬁcant in solutions and polymer blends,
and, in most cases, show diffusion-like behavior:
I(Q,t)∝π
δε(Q,t)δε(−Q,t)Δ∝S(Q)exp(−DQ
2
t). (1.18)
Light scattering is often used for the analysis of diffusion coefﬁcients and/or hy-
drodynamic radii of synthetic and biologicalmolecules in solution. The traditional
technique for this kind of study is photon-correlation spectroscopy [48]. However,
this technique does not analyzeI(Q,t), but the intensity–intensity correlation
function:
g
2
(Q,τ)=
πI(Q,t)I(Q,t+
τΔ
πI(t)
2
Δ
. (1.19)
As a result, the measured PCS correlation time is half that of the real relaxation time
ofI(Q,t).
(b) Molecular approach
The second approach to describe light scattering is based on molecular polarizabil-
ity. In this case,
δεyz(r,t)is expressed in terms of an optical polarizability tensor
αyz(i,t),whereiis the index of the molecule:
I
yz(Q,ω)=
1
2π
π
∞
−∞
exp(−i ωt)dt
∑
i,j
αyz(i,t)αyz(j,0)exp[iQ{r i(t)−r j(0)}]

.
(1.20)
The correlation function inπΔ, as usual, can be decomposed into “self”(i=j)and
“distinct”(iλ =j)correlation functions. In the particular case of polarized(y=z)
scattering by a spherical molecule (assuming
αzz(i,t)=α):
I
zz(Q,ω)=
α
2
2π
π
∞
−∞
exp(−i ωt)dt
∑
i,j
exp[iQ{r i(t)−r j(0)}]

= α
2
S(Q,ω).
(1.21)
So, we again have a dynamical structure factor analogous to that obtained from
neutron scattering, but weighted by the optical polarizability of the molecule. The
latter depends on the molecular environment, density, etc. and thus the analysis is not
as straightforward as in the case of neutronscattering where the nuclear scattering
cross-section is the property of thenucleus independent of its surroundings.

18 A.P. Sokolov and V. Garc ´ıa Sakai
Anisotropy of the optical polarizability of the molecules gives rise to depolarized
light scattering(y =z). In the case of macromolecules, for example, optical
anisotropy arises because of anisotropy ofthe molecular structural units (e.g., a
monomer) and because of the nonspherical shape of the entire molecule (e.g., pro-
tein). In the ﬁrst case, one can measure the internal dynamics of the macromolecule,
whereas in the second case, motions of the entire molecule, such as a rotation, can
be studied.
To conclude this section, we want to emphasize that light scattering is the
most commonly used scattering technique for studies of the dynamics of soft
materials. As we showed above, it provides very similar information to that from
neutron scattering, despite the more complex and less straightforward microscopic
interpretation. However, it is limited by the rather lowQ-range coverage as was
shown in Fig.1.2. As a result, information on the geometry of the motions on a
molecular scale is not accessible. Nevertheless, it is very useful to have preliminary
light scattering data before performing neutron scattering experiments. They serve
as a good guide for the design of appropriate neutron scattering experiments since
they provide information about the frequency and temperature range of interest,
making the neutron experiment more efﬁcient.
1.7 X-Ray Scattering
IXS is a direct analog to light scattering (X-rays are very short wavelength light).
However, it has a tremendous advantage because the wavelength of X-rays is com-
parable to the characteristic interatomic and intermolecular distances. Moreover,
X-rays scatter essentially from atoms (although, to be precise, they scatter from
charges) and provide direct information on atomic motions. Recent progress in
synchrotron radiation has led to facilities that provide intense monochromatic and
coherent X-ray beams with very high photon ﬂux. These advantages make X-ray
spectroscopy very attractive for studying the dynamics of soft materials. In addition,
this technique requires only a small amount of sample and spectra can usually
be measured in relatively short time (at least, in comparison to neutron scattering
spectroscopy).
There are, however, signiﬁcant drawbacks to using X-rays. One of the main
problems is the high energy of the X-ray photons which is typically in the 10–40 keV
range. For comparison, neutron energies used for spectroscopy are in the meV
range, i.e., a million times lower. The high energy of the photons has two negative
implications. First, it is difﬁcult to achieve good energy resolution. Even the best
spectrometers can provide energy resolutions of only∼10
−7
of the incident energy,
i.e.,∼1meV(∼240GHz,∼1ps), which is not sufﬁcient for most of the studies
of soft materials. This is currently the highest resolution achievable [17]. Secondly,
X-rays of such energies can cause damage to samples since they are high enough to
break bonds and alter the properties of biological and synthetic macromolecules.
As a result, in many cases samples cannot be exposed to the X-ray beam for

1 Experimental Techniques for Studies of Dynamics in Soft Materials 19
-10-8-6-4-20246810
0
50
100
150
200
X-ray [cts/s]
IXS
E [meV]
0
10
20
30
40
50
INS
Neutrons [cts/h]
Fig. 1.5Comparison of inelastic X-ray (open circle) and neutron (open triangle) scattering spectra
measured from a sample of polybutadiene atT=140 K. Thedash lineshows the ﬁt to the damped
harmonic oscillator model and thesolid linepresents the inelastic peaks obtained from the ﬁt.
Data are taken from [50]. Note that, because of the different elemental scattering cross-sections for
neutrons and X-rays, the spectra should not be identical
a long time. To overcome this, researchers either limit the measurement time or
constantly move the sample (to illuminate different parts) during the measurements.
This imposes additional limitations on the experiments that can be performed with
X-ray spectroscopy.
There are now two well-developed X-ray scattering techniques that have already
found broad applications in studies of dynamics in soft materials. The ﬁrst one
ishigh resolution inelastic X-ray scattering (IXS)[17,49,50]. In some sense,
it is analogous to light Brillouin scattering, but covers a much higherQ-range. It
is the only direct way to measure the bulkmechanical properties of materials at
the nanoscale. IXS provides the same information as inelastic neutron scattering
(INS), and despite having poorer resolution, measurement statistics are much better
(Fig.1.5shows hundreds of counts per second on the X-ray scale compared with
only tens of counts per hour on the neutronscale). In addition, in contrast to
neutrons, there are no kinematic limitations. (One can only measure Brillouin
scattering using radiation that propagates faster than the speed of sound in the
material being analyzed; for neutrons, this can mean working at unfeasibly small
scattering angles or with insufﬁcient energy resolution.)
Many interesting results from studies of vibrational dynamics in glass-forming
systems have been achieved using IXS [17,49]. These results reveal that the
broadening of the IXS modes varies essentially withQ
2
, and in most cases, is
temperature-independent [49]. This unusualQ-dependence (traditional hydrody-
namic predicts aQ
4
dependence) remains a subject of active discussion. IXS also
provides interesting microscopic information on the nature of boson peak vibrations,

20 A.P. Sokolov and V. Garc ´ıa Sakai
which are low-frequency modes that are general for the dynamics of small molecular
glass-forming systems, polymers, and biological macromolecules. Unfortunately,
the limited energy resolution prevents further signiﬁcant progress in this direction.
There are currently attempts to move to longer wavelengths (deep UV light) using
synchrotron radiation in order to improve the energy resolution [51].
The other important technique isX-ray photon correlation spectroscopy (XPCS)
[52,53]. This is a complete analogy to usual light PCS, only using short wavelength
photons. As a result, it provides the same information on the dynamics of soft
materials, but on molecular length scales(Fig. 2.2). It has been demonstrated that
XPCS can be used effectively for studies of thin polymer ﬁlms and polymeric
nanocomposite materials [52,53]. As for X-rays in general, but especially for XPCS,
the drawback is the radiation damage caused to samples. As mentioned before,
samples should either be exposed to the X-ray beam for just a few minutes or have
to be in constant motion during the measurements. This results in the limited time
range of the dynamics that can be measured currently with XPCS, those slower
than∼10
−5
–10
−3
s. Future construction of X-ray sources with even higher brilliance
might shift this limit to even shorter times [54].
1.8 Concluding Remarks
In this chapter, we have presented an overview of the major experimental techniques
traditionally employed for studies of dynamics in soft materials. Most of these are
techniques that can be placed in any research laboratory (e.g., university laboratory).
Unfortunately this is not the case with neutron scattering spectroscopy. In this case,
the time gap between generating an idea and performing the experiment might be as
long as 1 year. In addition, neutron scattering experiments suffer from a rather weak
neutron ﬂux per spectrometer in comparison to, for example, X-ray scattering.
Despite these disadvantages, neutron scattering spectroscopy has unique prop-
erties that make it very attractive for studies of dynamics in soft materials. The
most important advantage is that it measures atomic motions directly, on an absolute
cross-section scale, because neutrons scatter directly from nuclei. Thus, the results
of neutron scattering experiments can be compared directly and quantitatively to
model predictions and to results of computer simulations. The wavelength range
of neutrons,
λ∼0.1–1nm, opens the scattering wavevector range to one that
probes interatomic and molecular distances. Thus, it is a powerful technique to
probe microscopic details of molecular motions (diffusive-like, rotation, etc.), their
geometry and length scale, in addition to characteristic frequencies and relaxation
times. The frequency (time) window accessible to neutron scattering is much
broader than that accessible to X-rays (Fig.1.2) and neutron beams do not damage
samples, even those as sensitive as biological systems. Furthermore, most soft
materials contain hydrogen atoms and, given the signiﬁcant difference in scattering
intensity from hydrogen and deuterium, isotopic substitution can be used to study
the dynamics of a speciﬁcally labeled part of a molecule or a particular system

1 Experimental Techniques for Studies of Dynamics in Soft Materials 21
component. Coherent neutron scattering also provides important information on
cooperativity (coherency) of atomic motions. This advantage, however, has not yet
fully exploited in the soft matter ﬁeld.
All of these advantages make neutron scattering spectroscopy a unique tool
for studies of dynamics in soft materials that cannot be matched or substituted
by any other current experimental technique. In our personal view, one of the
main obstacles to a wider and more efﬁcient use of neutron scattering is the
absence of a means to focus neutron beams to small enough spots (e.g., mm-size
spots while maintaining divergence and hence resolution). As a result, neutron
spectroscopy requires large amounts of sample which are unfeasible for many bio-
and nanotechnologies. Moreover, to achieve high enough signals, experimentalists
have to sometimes work on the border of multiple-scattering effects which makes
the data analysis harder; this approach is never used in, for example, light scattering,
simply because there are enough incomingphotons. Therefore, the ability to create
a higher neutron ﬂux per square millimeter (not even the total, just per surface area
of the sample) without losing important characteristics of the neutron beam would
signiﬁcantly broaden the use of this technique, improve signal-to-noise ratio, and
allow measurements with negligible multiple-scattering contributions.
To conclude this chapter, we want to emphasize that the use of neutron
scattering spectroscopy is most efﬁcient when combined with other complimentary
techniques. For example, dielectric spectroscopy provides accurate temperature
dependencies, spectral shapes, which can be fed back into the analysis of neutron
scattering spectra and provide accuratemicroscopic information on underlying
molecular motions. The use of light scattering and molecular dynamics simulations
as a guide to the accurate planning of a neutron scattering experiment is also very
efﬁcient. The combination of different experimental techniques is always the best
strategy!
AcknowledgmentsThis work was sponsored by the Division of Materials Sciences and
Engineering, DOE Ofﬁce of Basic Energy Sciences.
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Chapter 2
Computational Tools to Understand Inelastic
and Quasielastic Neutron Scattering Data
Mark R. Johnson, Miguel A. Gonz´alez, Mohamed Zbiri, and Eric Pellegrini
2.1 Introduction
Neutron scattering (NS) probes the way atoms move, giving an experimental
signature of the dynamics. According to Newton’s laws of motion, an atom moves
because a force has been exerted on it and it subsequently ﬂies ballistically. In
condensed matter, due to the high atomic density and the bonding between atoms,
the ballistic regime is short and atoms are continually experiencing new forces
and accelerations. The accelerationadepends linearly on the forceFwhere the
proportionality constant is the massmof the atom:
F=ma. (2.1)
Ignoring zero point motion, there is no motion when all atoms are at minima of
the potential energy surface (PES) and this arrangement of atoms constitutes an
equilibrium structure. The most stable structure is the one with the lowest total
energy, i.e. the global minimum of the PES.
The equilibrium structure is the starting point for lattice dynamics (LD) calcu-
lations. Vibrational motion measured by inelastic neutron scattering (INS) explores
the PES in the vicinity of the minima. Since the PES depends on all the atomic
coordinates, displacinga single atom slightly from equilibrium will (a) induce a
restoring force on that atom and (b) induce inter-atomic forces on all other atoms.
In the case of a harmonic potential energy well, the magnitude of the restoring
force depends linearly on the amplitude ofthe displacement. In real systems the
harmonic approximation is valid for small displacements (e.g.<0.05˚A). The inter-
atomic forces, measured by displacing atoms from equilibrium positions, are used
to construct the dynamical matrix (DM), which is an expression of the equations of
M.R. Johnson () • M.A. Gonz´alez • M. Zbiri • E. Pellegrini
Institut Laue Langevin, BP 156, 6 rue Jules Horowitz, 38042 Grenoble, France
e-mail:[email protected]
V. G a r c´ıaSakaietal.(eds.),Dynamics of Soft Matter: Neutron Applications,Neutron
Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-02,
© Springer Science+Business Media, LLC 2012
25

26 M.R. Johnson et al.
motion for all atoms. Diagonalising the DM gives a set of normal modes (NM) or
harmonic oscillators, each one having a characteristic frequency and a displacement
vector describing the relative motion of all atoms in that mode. The NM can be
used to construct NS observables like the generalised vibrational density of states
(GVDOS) and the scattering functionS(Q,
ω)for powders or single crystals. In
the case of an equilibrium structure, well-separated in coordinate space from other
minima in the PES, the NM frequencies will all be bigger than or equal to zero.
A larger atomic displacement, may however, push the structure towards another
minimum of the PES, there being no restoring force on the displaced atom. In this
case, the LD calculation will result in negative frequencies (or rather imaginary
frequencies, see later) which are a signature of the instability.
The equilibrium structure is also the starting point for a molecular dynamics
(MD) simulation. In a MD simulation, the NS observables mentioned above
are extracted from the time-dependent trajectories of the atoms via correlation
functions, like the velocity auto-correlation function and the van Hove correlation
functions. The system has to be displaced from equilibrium by an initial “impulse”,
which is achieved with a Boltzmann distribution of initial velocities corresponding
to the required simulation temperature, and thereafter the motion is perpetual. Each
structure in the series of the trajectory of the MD simulation is calculated from the
previous structure using the instantaneous forces and therefore the accelerations
acting on each atom.
In a MD simulation, the system will vibrate about its equilibrium geometry,
unless the kinetic energy is sufﬁcient to allow atoms to cross maxima in the PES,
giving rise to large amplitude, atomic displacements. This motion can be local, for
example rotations of molecular groups, or long-range, giving rising to liquid-like,
translational diffusion. The experimental signature is quasielastic neutron scattering
(QENS), which is a broadening of the elastic peak, as opposed to discrete, inelastic
peaks. Whereas LD calculations correspond to a temperature of zero Kelvin, since
the atomic positions are optimised to minima of the PES, MD simulations introduce
temperature through the kinetic energy of the atoms.
Forces are therefore central to LD and MD methods. They are calculated from
an expression of the total energy of the system as the derivatives with respect to the
appropriate atomic coordinates. While there are many methods and approximations
used for calculating the energy of a system of atoms, two are widely available and
enable a wide range of science and systems to be tackled. The energy of a set
of atoms depends on how the valence electrons of the atoms organise themselves
around the nuclei and the core electrons. A correct description of the electronic
structure requires Schroedinger’s equation to be solved for the system of electrons
and nuclei. Using traditional, Hartree-Fock methods [41,42], these calculations
can scale as badly as the seventh power of the number of atoms, i.e. doubling
the number of atoms would increase the computational cost by a factor of 128
(N
7
whereNis the number of atoms). One of the most computationally efﬁcient
methods is to calculate the one-electron density ﬁeld around the nuclei rather than
the many-electron wavefunction. This approach is embodied in density functional
theory (DFT), which recovers the missing correlation in the Hartree-Fock approach,

2 Computational Tools to Understand Inelastic and Quasielastic... 27
and codes based on this theory typically offerN
3
scaling. DFT-based methods can
be applied to aperiodic (e.g. molecules) and periodic systems of hundreds of atoms
and a timescale up to 100 ps can be reached, therefore enabling a wide range of
materials studied by NS to be simulated accurately. The scope of DFT is further
enhanced by linear scaling methods which allow 10
4
atoms to be handled, thereby
increasing the overlap with classical methods.
Larger systems like polymers and bio-molecules (the principal subject of this
book) for which, in addition, slower dynamicsand therefore longer timescales are
of interest, generally require methods in which the computational cost per atom is
much lower. This is achieved by treating electrons implicitly. The chemical bonds
that are formed by electrons are described by springs. The net electron density
around atoms is described by point charges. The long-range correlation of electrons
(not treated in standard DFT-based methods) is described by the dispersive term in
the Van der Waals interaction, which is typically proportional tor
−6
,whereris the
inter-atomic distance. Springs, point charges and interactions like electrostatic and
VDW interactions constitute a force ﬁeld (FF), which is a parameterized, analytical
expression for the energy of a system. The complexity of the FF (the number of
springs and interactions represented) and the parameters determine the precision and
the extent to which a FF can be used on a range of systems. FF-based methods allow
10
5
atoms and a timescale up to 10 ns to be treated easily, with additional orders of
magnitude in size and/or time available on more powerful computational resources.
For larger systems, when atomic details are no longer relevant, for either the science
under consideration or for the stability of the simulation, uniﬁed atom and coarse-
graining techniques allow the time and length scales to be further increased.
LD and MD simulations can equally well be performed with electronic structure
or classical, FF-based energy calculations. The combination of simulation method
and energy calculation depends on the physical effects that are sought. LD offers the
most detailed information of how condensed matter systems vibrate, but this motion
is described within the harmonic approximation for small amplitude displacements
about an equilibrium structure. Large amplitude motion like diffusion at ﬁnite
temperature requires MD simulations to be performed. If the large amplitude motion
involves bond breaking, like proton migration in a hydrogen bond, then electronic
structure-based energy calculations have to be used. Otherwise the size of the system
and the (resulting) time scale of the associated dynamics dictate which type of
energy calculation has to be used. Often, LD is used with DFT and MD is used
with FFs, the second combination being common for soft matter studies. But this
is an over-simpliﬁcation, as we will demonstrate, as the scope of DFT increases
from treating tens of atoms in the past to hundreds of atoms today and thousands of
atoms in the very near future. In addition,increased computing resources enable ever
larger matrices to be diagonalised, extending the scope of LD methods. Principal
component analysis bridges the gap between LD and MD by enabling approximate
NMs to be extracted from MD simulations.
This chapter expands on the above ideas, illustrating them with recent examples,
from both hard and soft matter. The chapter is presented in sections as follows.
Section 2.2 relates total energy to forces acting on atoms, starting from the simplest

28 M.R. Johnson et al.
example of a harmonic oscillator. Section 2.3 introduces LD starting from the
equations of motion for a linear arrangement of beads. In Sect. 2.4, an overview of
molecular dynamics is presented. LD and MD approaches are compared in Sect. 2.5.
Total energy calculations are described for DFT and FF methods in Sect. 2.6, high-
lighting the increasing scope of DFT linear scaling methods. Section 2.7 presents a
discussion of the relative merits of FF and DFT and the way in which these methods
can be combined is highlighted. We have chosen to present simulation methods, LD
and MD, before the energy calculations onwhich they are based because, to a certain
extent, the choice when starting simulations should be made in this order, motivated
by the physical phenomena of interest. We hope also, in this way, to decouple
methods and energy calculations, avoiding established associations like “DFT
can only be used for LD”. Section 2.8 discusses how to compare simulated and
experimental data concerning, in particular, how the instrument resolution function
and other artefacts can be included in thecomparison. The chapter concludes with
perspectives for the future. The basis for concepts and methods in this chapter
originates in part in the ﬁeld of hard matter, particularly for LD and DFT. Where
appropriate, the extension to soft matter is discussed. A wide range of publications
exists on simulations and we recommend the following books [1–4] and articles
[5,6] for those who seek more information after reading this chapter.
2.2 Energy to Forces
One of the simplest, most pertinent examples of a potential energy well is the
harmonic potential, described by aquadratic dependence of the energyEon the
displacementx.
E=
kx
2
2
. (2.2)
The general expression for the forceFacting on a particle in this well is,
F=−
dE
dx
=−kx, (2.3)
which allows the force constantkto be deﬁned.
k=−
dF
dx
=
d
2
E
dx
2
. (2.4)
The equilibrium position of a particle in this well isx=0, where the energy is a
minimum and the force acting on the particle is zero. If the particle is displaced
from equilibrium, it will oscillate perpetually with a frequency
ωwhere
ω
2
=
k
m
, (2.5)
andmis the particle mass.

2 Computational Tools to Understand Inelastic and Quasielastic... 29
This simple development allows us to introduce the “mass effect” which is used
to interpret the vibrations of a material when one of the atom types (and possibly
just one site) is replaced by another atom type. If the two materials are iso-structural
and iso-electronic (e.g. in the case of isotopic substitution, like deuteration), then
the “mass effect” on the frequency is
ω
ω
ω
ω
π
2
=
m
ω
m
, (2.6)
since the PES, described here byk, is unchanged.
The harmonic potential is so useful because it is the ﬁrst non-zero term in the
expansion of the PES about the equilibriumposition, the linear term being zero at
the equilibrium position. It transpires, in practice, that providing the temperature of
a crystal is well below the melting point, the harmonic approximation is good.
2.3 Lattice Dynamics and INS
In text books, LD are typically developed in terms of a linear chain of beads. For
a monatomic chain, the only vibration is an acoustic phonon (see below). The next
level of complexity involves two atoms of different mass,mandM, in a periodic
cell of lengtha, so the distance between atoms isa/2(seeFig.2.1).
The equations of motion for the two distinct particles are developed by combin-
ing foregoing expressions for the forceF.
F=ma=−kx (2.7)
m¨x
mn=k[x Mn+x
M(n−1) −2xmn] (2.8)
M¨x
Mn=k[x mn+x
m(n+1) −2xMn]. (2.9)
a
unit cell ‘n’ unit cell ‘n+1’unit cell ‘n-1’
Fig. 2.1Diatomic chain of massesm(red)andM(yellow) and periodic cell lengtha

30 M.R. Johnson et al.
ω
Optic branch
Acoustic branch
0 π/a−π/a
q
Fig. 2.2Dispersion curves for the diatomic chain
Assuming a plane wave solution for the periodic system of the form
x
mn(t)=A me
i(qna− ωt)
(2.10)
x
Mn(t)=A Me
i(qna+qa/2− ωt)
, (2.11)
gives a pair of simultaneous equations that can be expressed in determinant form.
4
λ
λ
λ
λ
λ
2k−M
ω
2
−2kcos
∑
qa
2
σ
−2kcos
∑
qa
2
σ
2k−m
ω
2
λ
λ
λ
λ
λ
=0. (2.12)
qis the wavevector corresponding to a wavelength
λ=2π/q. Solving the equations
gives two solutions which depend onq, as shown in Fig.2.2. The dependence of
qon
ωis called dispersion. The lowest frequency dispersion curve is the acoustic
branch for which the frequency
ωis zero atq=0. The wavelength of this mode
is inﬁnite, which means all atoms move in phase and there is no restoring forcing
(k=0). A restoring force arises when there is relative motion of atoms, for example
whenq=
π/a, which describes the shortest wavelength vibration in which only the
heavy atoms move with frequency
ω
2
=2k/M.q=0 is called the Brillouin zone
centre, or the Gamma point, andq=
π/ais the zone boundary. Extendingqbeyond
the zone boundary (by conventionqbecomesQ) reproduces the dispersion of the
ﬁrst zone, so no new information is obtained, although theseQvalues are accessible
in experiment and they determine, in part, the spectral intensity.
The higher frequency dispersion curve is called the optic branch. At the zone
centre, light and heavy atoms move in opposite directions and the centre of mass is
stationary. At the zone boundary, only light atoms move with frequency
ω
2
=2k/m.

2 Computational Tools to Understand Inelastic and Quasielastic... 31
In terms of simulation methods that will be presented below, it is important
to note that a periodic model of lengthais able to describe vibrations of all
wavelengths. This is because the interaction range isa/2, so the inter-atomic
interactions are contained within the periodic cell. In this model, long wavelength
vibrations depend only on nearest neighbour interactions.
For pedagogical purposes (no new information is obtained), the supercell can be
extended to a length of 2aand the determinant can be expressed as
λ
λ
λ
λ
λ
λ
λ
λ
2k−Mw
2
−ke
iqa/2
0 −ke
−iqa/2
−ke
−iqa/2
2k−mw
2
−ke
iqa/2
0
0 −ke
−iqa/2
2k−Mw
2
−ke
iqa/2
−ke
iqa/2
0 −ke
iqa/2
2k−mw
2
λ
λ
λ
λ
λ
λ
λ
λ
=0, (2.13)
showing the general form of the elements in the determinant. New information
would be obtained if the extended cell was used to include second neighbour
interactions, which remove the zero elements of the matrix. Generalising (2.13)
further to three dimensions, the DM has matrix elements of the form,
DM
ij=
k
ij
√
mimj
e
i[q·(ri−rj)]
, (2.14)
where the indices{i,j}run over theNatoms, massm
i, and their Cartesian
coordinatesr
i={x,y,z}. The matrixk ijis called the Hessian.
Diagonalising the DM gives 3Nq-dependent frequency branches which are
composed of 3 acoustic branches and 3N-3 optic branches. The eigenvalues of the
DM are
ω
2
. The corresponding eigenvectors describe the relative displacements
of the atoms/particles, the physical displacements being obtained by weighting
by the mass, i.e. dividing each eigenvector component by the square-root of the
corresponding atomic mass.
2.4 Direct Way for Calculating Phonons
A general and conceptually simple method for calculating phonons is the direct
method or supercell approach, which entails a complete determination of the DM.
This approach is implemented in the PHONON code [7] used at ILL, France, and
a related code, NMScatt [8], developed for bio-molecular systems [9]. The steps of
the direct method are as follows:
1. Determine the equilibrium geometry of the periodic system (e.g. crystal cell).
2. Construct a supercell that will include all interatomic interactions (ideally a cube
of side∼10˚A). For soft matter systems, this condition will always be satisﬁed.
When FF are used for the underlying energy and force calculations, the supercell
side should be twice the cut-off for long-range VDW and Coulomb interactions.

32 M.R. Johnson et al.
3. Generate a set of structures in which each crystallographically inequivalent
atom is displaced along the inequivalent Cartesian directions from equilibrium
(typically 3ndisplacements wherenis the number of inequivalent atoms)
4. For each structure, calculate the inter-atomic forces.
5. Construct the DM from these forces, the amplitude of displacement and
symmetry.
6. Diagonalise the DM for any value ofq.
All the vibration frequencies obtained this way should be bigger than or equal to
zero. Negative frequencies are actually imaginary frequencies since the eigenvalues
of theDMare
ω
2
and these are indicative of physical or numerical problems, as
follows;
The equilibrium structure is metastable and an atomic displacement drives the
system towards another, stable structure.
The equilibrium structure is not well-enoughdetermined. Residual forces should be
<0.01eV/˚A when using DFT. With a FF, which is an analytical expression for
the total energy, residual forces are several orders of magnitude smaller.
The atomic displacement is not big enough toinduce forces that are bigger than the
residual forces. In DFT-based calculations, the atomic displacement is typically
0.05˚A and the on-site restoring force should be∼0.5eV/˚A. For FF-based
calculations, the atomic displacementcan be orders or magnitude smaller since
the residual forces are very close to zero. This fact removes the problem in the
ﬁrst point above for FF-based calculations.
The supercell does not contain all inter-atomic interactions, which may be the
case in DFT-based calculations if electrostatic interactions are important. In
FF-based calculations, the supercell should be matched to the cut-off for VDW
and Coulomb interactions (see above).
2.4.1 Analysis: Extracting INS Observables
The vibration frequencies from the DM can beused to calculate dispersion curves,
the total vibrational density of states and related thermodynamic quantities, like heat
capacity, entropy, free energy, etc. The displacement vectors are used to determine
the partial densities of states (pVDOS), for example per atom type, from which the
GVDOS, the experimental observable, can be calculated.
GVDOS=
∑
i
σi
mi
pVDOS
i
(2.15)
pVDOS
i
=∑
j
e
2
i,j
. (2.16)
σandmare the atomic scattering cross-section and mass, respectively, ande i,jare
thejcomponents of the eigenvectors for atom typei. The eigenvectors are also used
to determine spectral intensity, be it for coherent scattering along dispersion curves
or for incoherent scattering from oriented samples [10]orpowders[11].

2 Computational Tools to Understand Inelastic and Quasielastic... 33
Fig. 2.3Dispersion curves along high symmetry directions for a pyrochlore structure of strontium
gallium oxide(SrGa
12O19)
Figure2.3shows a set of calculated dispersion curves for a metal oxide [12],
which are often shown in black and white. The grayscale here indicates the mode
intensities for coherent scattering thatare determined from the dynamic structure
factor, rather than (2.15)and(2.16). It reveals that modes at the M and Y points
at 6 meV(∼1.5THz)would dominate an INS measurement. The corresponding
atomic displacments drive relaxation in this system. Gamma point modes at this
energy transfer, which could be measured by IR/Raman, have different atomic
displacements, and they cannot drive the relaxation process in question. In this
case, calculations help to reconcile different spectroscopic observations and give
an accurate description of the active modes.
By deﬁnition, a stable LD calculation can only describe inelastic scattering, since
the displacements about the equilibrium structure are small. MD techniques (see
below) give access to both quasielastic and inelastic scattering.
2.4.2 Powder-Averaged Lattice Dynamics
Most samples studied by INS are powders. When the GVDOS is extracted from INS
data, this is typically done in the incoherent approximation in which the inelastic
intensity is assumed to increase smoothly withQ
2
. But for samples with atoms that
are mainly coherent scatterers,S(Q,
ω)does not increase smoothly withQ
2
and
the variations in intensity contain information about the dispersion and therefore

34 M.R. Johnson et al.
Fig. 2.4CalculatedS(Q,w)map (on a logarithmic intensity scale) for a skutterudite in which La
guest atoms are encaged in a Fe-Sb network. The verticalQ-axis extends from 0 to 5˚A
−1
.The
horizontal, energy axis extends from 0 to 35 meV
inter-atomic coupling, that would normally be obtained from single crystal measure-
ments. Thus in a recent development, powder-averaged lattice dynamics (PALD)
have been performed in which the coherent structure factor has been determined for
a complete set of randomly chosen points in Brillouin zones spanning theQ-range
of interest [13]. This is in contrast to typical calculations performed for crystals in
which the intensity variation along dispersion curves in high symmetry directions
is of interest (see above). Figure2.4shows a calculatedS(Q,
ω)map for a powder
skutterudite sample [14]. Comparison of this numerical data with experimental data
from a time-of-ﬂight spectrometer enabled the underlying harmonic LD calculation
to be validated which, in turn, disproved a model of “rattling modes” in these
potential thermoelectric materials.
2.4.3 Analysing Large-Scale Phonon Calculations
At the heart of the LD approach is the diagonalisation of the DM. This can be
performed exactly for very large matrices (>10
8
elements) allowing complex sys-
tems to be treated in this way. However, the number of eigenmodes, matched by the
dimension of the corresponding, atomistic eigenvectors, can no longer be analysed
in a meaningful way by the methods used in smaller, traditionally hard matter
systems, for example simple visualisation of modes and their symmetry analysis.
Recent calculations on DNA have highlighted this problem [15]. The advantage of

2 Computational Tools to Understand Inelastic and Quasielastic... 35
atomistic eigenvectors is that they can then be analysed on different length scales
by summing the atomic displacements over user-deﬁned beads and then projecting
these bead displacements onto appropriatevectors. This approach was used to
explore the base-pair opening character of vibrational modes, by projecting bead
displacements onto the base-pair vector, see Fig.2.5. By deﬁning nucleotides as
beads, the base-pair opening character peaks at 12–15 meV, well below the energy
corresponding to the melting of DNA (T
m≈370K i.e.∼32meV). When base
molecules are deﬁned as beads, the maximum value of base-pair opening extends up
to the energy of melting, showing that this type of vibrational mode can drive DNA
melting and related processes. The lower frequency modes are delocalised over 4–5
base-pairs, whereas the higher frequency modes tend to be localised on a couple of
base-pairs.
2.5 Molecular Dynamics for INS and QENS
A MD simulation describes a perpetual, time- and temperature-dependent evolution
of a system. The MD trajectory is a set of structures in which each atom has a
positionrand velocityv. The next structure in the trajectory is calculated from the
instantaneous forcesFacting on the atoms which gives the acceleration from (2.1).
a=
F
m
. (2.17)
According to the simplest integration of the equations of motion, the positions of
the atoms in the next structure are
r(t+dt)=r(t)+v(t)dt+
a
2
dt
2
, (2.18)
and their velocities are
v(t+dt)=v(t)+a(t)dt, (2.19)
wherer,v,andaare vectors describing position, velocity and acceleration respec-
tively. dtis the simulation time step which should be signiﬁcantly(×5)shorter than
the period of the highest frequency vibration. For systems containing H in which
the highest frequency vibrations are∼400meV,dtis typically 1 fs. For systems
composed of heavier atoms, or molecular systems treated as full or partially rigid
units, dtcan be as long as 5 fs.
The total simulated time is equal to dttimes the number of simulation steps, so
the longer dt, the slower the dynamics that can be investigated (for the same number
of steps). dtis, however, a time in which atoms ﬂy ballistically, and if dtis too long,
inter-atomic distances can become very short, forces and accelerations very large,
and the simulation unstable. A long time step may also lead to problems of energy
conservation during MD, which must be checked.

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I Padovani, che avevano ragione di temere che Cane in virtù del suo
titolo di vicario imperiale nella Marca Trivigiana, non pretendesse di
avere sopra la loro città que' medesimi diritti ch'esercitava sopra
Vicenza, più non ascoltando che la loro impazienza e la loro collera,
armarono le loro milizie ed assoldarono mercenarj per intraprendere
la guerra. La gioventù aveva piacere che incominciasse: stanca della
monotonia della pace, di cui godeva da tanto tempo la sua patria.
«Pure, dice Ferreto di Vicenza, quando la guerra fu intimata dai due
popoli, gli abitanti delle campagne furono i primi ad essere attaccati:
il primo segno delle ostilità fu la rapina delle loro gregge e de' loro
mobili. I contadini che in questo subito attacco non furono fatti
prigionieri, sforzaronsi di condurre in città e di deporre in luogo
sicuro tutto quanto poteva essere trasportato. Allora si videro gli
agricoltori condurre un lungo ordine di carri, sui quali avevano
frettolosamente caricati i rustici loro mobili e i vasi delle loro cantine;
mentre le madri, coi loro fanciulli al seno o sopra le spalle, venivano
a dormire sotto gli stessi portici delle nostre case. Questa maniera di
guerreggiare, di uccidere e far prigionieri i cittadini, di rubare i loro
beni, e di bruciarne le case, veniva a noi insegnato dagli stranieri
mercenarj che avevano sempre vissuto nei campi. Quante volte non
abbiamo noi veduto strascinarsi da questi empi soldati, che Cane
pagava a prezzo d'oro, truppe di contadini padovani colle mani
legate alle reni? Essi custodivano questi prigionieri nella nostra patria
e crudelmente li maltrattavano per obbligarli a riscattarsi. Nè i
mercenarj di Padova trattavano più dolcemente i contadini di
Vicenza: come mai quest'infelici avevano meritate tante
ingiurie!»
[323].
La prima conseguenza della guerra fu l'aggravarsi della tirannia di
Cane sui Vicentini: quattro gentiluomini furono da lui incaricati
dell'assoluto governo di questa città; e perchè più prontamente
potessero percepire le imposte, tutte le immunità del popolo, tutte le
leggi furono abolite. Allora scoppiarono in Vicenza delle congiure
contro Cane, le quali giustificarono in certo modo le criminali
inquisizioni, l'esilio, la confisca dei beni di una parte della nobiltà che
rifugiossi in Padova, e che dopo tale epoca portò poi le armi contro

la patria. Ma la libertà non era meno in pericolo a Padova, ove ogni
zuffa era cagione di nuove animosità contro i Ghibellini: il loro capo
Guglielmo Novello, attaccato dai sediziosi nel palazzo pubblico fu
massacrato innanzi allo stesso pretorio; e de' suoi partigiani alcuni
fuggirono, altri come nemici della patria furono condannati a
perpetuo bando
[324].
Il luogo in cui si veniva più frequentemente a battaglia tra i due
popoli, era quello in cui il Bacchiglione, fiume che attraversa il
Vicentino, si divide in due rami, uno de' quali, dirigendosi al Sud-
Ovest, bagna le campagne d'Este, e l'altro al Sud-Est quelle di
Padova. L'abbondanza delle acque raddoppia la fertilità di quelle
ricche campagne, ed il possesso del fiume per farne scendere una
minore o maggior parte dall'una o dall'altra parte era della più alta
importanza pei due popoli, i quali attaccarono, rovesciarono,
rialzarono più volte le dighe. In queste zuffe i Padovani erano
sempre superiori di numero e di ricchezze; ma Cane, la di cui armata
era quasi esclusivamente formata di mercenarj, accostumati fino
dalla fanciullezza al mestiere delle armi, e che non sapevano cosa
fossero la fatica o la pietà, vinceva i Padovani per conto della
disciplina e dell'arte militare.
Avendo i Padovani adunate le truppe sussidiarie di Cremona, di
Treviso, del marchese d'Este e gli esiliati di Vicenza e di Verona; ed
inoltre avendo assoldati alcuni condottieri, tra i quali distinguevansi
due Inglesi, Bertrando ed Ermanno Guglielmo
[325], formarono
un'armata di 10,000 cavalli e 40,000 fanti; armata formidabile che
pareva bastante a conquistare tutta la Lombardia. Pure sì grande
armata, invece di fare qualche strepitosa impresa, non giovò ad altro
che ad attirare sopra la Venezia un altro flagello. Si tenne lungo
tempo accampata, esposta all'ardore del sole, in riva a' fiumi, le di
cui torbide acque appena si muovono; le malattie vi presero piede,
ed una crudele epidemia distrusse nello stesso tempo i due campi e
le due città.
Il massacro di Guglielmo Novello di Campo Sampiero, e l'espulsione
de' Ghibellini suoi partigiani non riuscirono utili soltanto alla parte

guelfa, ma ancora alla fazione aristocratica che acquistò maggiore
influenza ne' consigli della repubblica. Pel corso di più di mezzo
secolo Padova erasi conservata fedele alla Chiesa, e l'aristocrazia
spalleggiava sempre il partito che una città aveva seguìto più lungo
tempo. Per altro i capi del governo non appartenevano ad antiche
famiglie: erano Pietro Alticlinio, avvocato, e Ronco Agolanti. Avevano
amendue ammassate grandi ricchezze coll'usura, e l'uno e l'altro
abusavano del credito che loro dava lo stato, in particolare
permettendo ai loro figli di valersene per soddisfare alle proprie
passioni. Amendue in onta al partito ghibellino, di cui avevano divise
le spoglie, ed in onta al popolo che avevano escluso dal governo,
non erano meno esosi alla casa dei Carrara, la più ricca della nobiltà,
la più popolare, e quella che colla sua ricchezza minacciava più delle
altre la libertà. Due giovanetti di questa casa, Nicola ed Obizzo,
eccitarono, contro il sentimento de' loro parenti, una sedizione per
disfarsi di questi due capi della repubblica. Introdussero moltissimi
contadini in città; ed incontrando Pietro Alticlinio sulla piazza del
mercato, gli furono addosso e lo sforzarono a fuggire. Nello stesso
tempo incominciarono a gridare, viva il popolo, viva il popolo solo!
Da tutte le bande si corse alle armi: invano il podestà co' suoi sgherri
occupò la piazza del pretorio, i sediziosi si attrupparono in tutte le
altre; invano, così consigliato dal vescovo, il podestà ordinò alle
compagnie della milizia di unirsi nella piazza grande per marciare di
là al proprio quartiere: si allontanarono a stento non più di cento
cinquanta passi e ben tosto tornarono a riempiere la maggior piazza.
Intanto i Carrara, ripetendo il grido di viva il popolo, vi aggiunsero
quello di morte ai traditori; ed i loro partigiani che si
frammischiavano in ogni gruppo di persone, andavano susurrando di
affidare ai Carrara la vendetta nazionale. Ben tosto fu per
acclamazione rimesso lo stendardo del popolo ad Obizzo dei Carrara;
e questi alla testa della plebaglia, ripetendo il grido di morte, si volse
alla casa di Pietro d'Alticlinio. La casa fu saccheggiata, ed il popolo,
ad un tempo credulo e furibondo, si figurò di avervi trovate le prove
de' più odiosi delitti che si attribuivano a Pietro ed a' suoi figliuoli:
prigioni ov'erano stati chiusi di nascosto i loro nemici; sepolcri nei
quali trovaronsi i cadaveri di coloro che avevano fatto perire; un

albergo dipendente da loro, nel quale i viaggiatori si uccidevano di
notte affinchè il proprietario ne acquistasse le spoglie; per ultimo
gl'indizj d'altri inauditi delitti e meno verosimili: tutte le quali accuse
furono confermate con asseveranza, siccome fatti indubitati
[326]. Il
primo giorno fu interamente consacrato al saccheggio di questa
potente casa. All'indomani fu denunciato al popolo Ronco Agolanti,
e, sorpreso nel luogo ov'erasi nascosto, fu massacrato ed il suo
cadavere strascinato in pezzi per le strade. Suo fratello non tardò a
provare la medesima sorte; le loro case e quelle ch'ebbero la
disgrazia di trovarsi vicine, furono saccheggiate, e la plebaglia avida
di bottino attaccò in appresso tutti coloro che gli si denunciavano
come amici delle prime vittime. Una voce propose di vendicarsi di
colui, il quale, preparando una nuova tariffa delle gabelle, voleva
impoverire il popolo con odiose contribuzioni. Quello che veniva in tal
modo indicato alla rabbia popolare era Albertino Mussato lo storico, il
quale, per far fronte alle spese della guerra, aveva proposta una
nuova tassa, che credeva più eguale, e stava formandone il catastro.
All'istante i sediziosi si precipitarono verso la sua casa la quale era
assai forte ed unita alle mura della città: ne furono chiuse le porte, e
mentre la furibonda plebe attaccava la muraglia. Mussato salì a
cavallo fuori della vicina porta, e fuggì a briglia sciolta verso Vico
d'Aggere, ove si pose in sicuro. La sua casa fu salvata dal saccheggio
perchè vennero proposte al popolo nuove vittime. Si seppe che
Pietro d'Alticlinio e i tre figliuoli eransi rifugiati nel vescovado; Pagano
della Torre, in allora vescovo di Padova, fu forzato a consegnarli alla
plebe, la quale soddisfatta del loro supplicio cominciò a
calmarsi
[327].
All'indomani, ch'era il primo giorno di maggio del 1314, gli anziani
della città, accompagnati dai tribuni, o gastaldioni, con gli stendardi
del comune e del popolo convocarono l'assemblea dei cittadini. In
questa fu risolto di mettere fine alle vendette; che gli attruppamenti
ed il grido di morte nelle strade sarebbero vietati; che si darebbe
opera a ristabilire la pace tra le famiglie, guarantendola coi
matrimonj; che il governo verrebbe affidato a dieciotto anziani,
secondo l'antica pratica; che sarebbero assistiti dai tribuni; e che la

repubblica continuerebbe a governarsi colla protezione e sotto il
nome di parte guelfa. Albertino Mussato fu richiamato e compensato
dal governo de' sofferti danni.
L'indisciplina dei campi non era minore della licenza della città: noi
siamo omai giunti a quegli sventurati tempi in cui la sorte della
guerra non dipendeva più dalle milizie nazionali, a que' tempi ne'
quali la sicurezza e l'onore dello stato venivano confidati a braccia
mercenarie e straniere. Ogni giorno i soldati arrogavansi nuovi
privilegi, ed aggravavano sui popoli i crudeli diritti della guerra; ed in
pari tempo ponevano in dimenticanza la disciplina, l'ubbidienza ed il
coraggio delle antiche repubbliche italiane.
Poco dopo la sedizione del mese di maggio, i Padovani, sotto la
condotta del loro podestà Ponzino Ponzoni cremonese, attaccarono
la stessa città di Vicenza. Cane della Scala erasene allontanato per
soccorrere Matteo Visconti. Il primo di settembre, all'ora de' vesperi,
Ponzino alla testa dell'armata padovana, d'un ragguardevole corpo di
mercenarj sotto gli ordini immediati di Vanne Scornazano e di mille
cinquecento carri destinati al trasporto delle bagaglie e delle armi
dell'infanteria pesante, prese la strada che da Padova conduce a
Vicenza. Queste due città non sono lontane che quindici miglia, ossia
cinque ore di marcia, di modo che l'adunanza de' carri che Ponzino
aveva fatta venti giorni prima, e col più grande segreto per questa
spedizione, ci dà la più straordinaria idea della maniera con cui
facevasi allora la guerra; e tale era la mollezza degli uomini d'armi,
che durante questa breve marcia notturna, la maggior parte avevano
deposte le armi sui carri che li seguivano
[328].
In sul far del giorno l'armata padovana giunse innanzi alle mura del
sobborgo di san Pietro a Vicenza, senza che la sua marcia fosse
stata annunziata da veruno esploratore: le guardie delle porte erano
addormentate; ed alcuni Padovani leggermente armati,
attraversando la fossa, si resero padroni dei ponti levatoj e gli
abbassarono prima che i Vicentini pensassero a difendersi. Le
guardie risvegliandosi, fuggirono in città e ne chiusero le porte; ed i
Padovani senza adoperare le armi rimasero padroni del sobborgo. Il

suono delle trombe e le grida di viva Padova! annunciarono questa
vittoria agli abitanti, i quali incerti della loro sorte, desiderosi di
tornare sotto l'amministrazione repubblicana de' loro padri,
desiderosi di scuotere il giogo di Cane, ma inquieti dell'abuso che
forse si farebbe del diritto della guerra, guardavano tremando i loro
vincitori. Ben tosto un proclama in nome di Ponzino Ponzoni stabilì la
pena di morte contro chiunque si rendesse colpevole di furto o di
morte: gli abitanti del sobborgo vi corrisposero con grida di gioja,
gridando ancor essi viva Padova! e le madri portando i fanciulli nelle
braccia sotto i portici, insegnavano loro a proferire questi due
vocaboli.
Frattanto i Vicentini, per meglio difendere il corpo della città,
tentarono d'incendiare le case del sobborgo più vicine alle mura; ed i
Padovani, non sapendo approfittare della loro vittoria, stabilirono il
loro campo duecento passi lontano dal preso sobborgo, di cui
affidarono la guardia a Vanne Scornazano ed a' suoi mercenarj: ma
appena giunsero al luogo in cui volevano fissare il campo, che lo
stesso Scornazano, sortendo dal sobborgo, si avanzò verso il podestà
Ponzino e Giacomo di Carrara, che stava co' principali capi
dell'armata: «E qual è, disse, cittadini di Padova, la vostra maniera di
fare la guerra? che significa quest'indulgenza pei vinti? voi non
sapete approfittare della vittoria, e la vostra pretesa dolcezza sarà da
tutto il mondo giudicata debolezza e pusillanimità. Quando le vostre
genti furono vinte si sottrassero alle ferite o alla morte? vi diedero
mai i vostri nemici l'esempio di questa indulgenza, o piuttosto di
questa viltà? Coi nemici accaniti non devesi risparmiare nè il ferro,
nè il fuoco, nè il saccheggio. Accordate ai vostri soldati il bottino del
sobborgo, altrimenti tra poco gli abitanti ben sapranno trafugare
tutte le loro ricchezze
[329].»
Ponzino ed i capi del popolo si rifiutarono a questa domanda; ma i
mercenarj non avevano aspettata la decisione del consiglio; ed il
saccheggio era già cominciato. Gli sventurati abitanti del sobborgo,
cui era stata guarentita la sicurezza, furono all'improvviso trattati con
tutto il rigore; e lo stesso Ponzino chiuse gli occhi sulla condotta de'
proprj satelliti che davano l'esempio di tutti i delitti. I mercenarj,

incaricati di custodire la porta che comunica colla città,
l'abbandonarono per ispargersi per le case, e ben tosto la ciurmaglia
del popolo padovano arrivò sollecitamente dal campo per dividere le
spoglie del sobborgo. Furono gettate ne' campi tutte le munizioni
che erano state portate sui carri che seguivano l'armata, onde
caricarli de' più preziosi effetti del bottino: nè i sacri vasi delle chiese,
nè le cose de' monasterj furono rispettate; e la brutalità de' soldati
espose agli ultimi oltraggi le spose e le figlie de' Vicentini, e perfino
le vergini consacrate a Dio
[330].
Frattanto, avanti l'ora terza del giorno, era stato dato avviso a Cane
della Scala, che trovavasi a Verona, della presa del sobborgo; e tosto
gittatosi in ispalla l'arco, ch'egli soleva spesso portare all'usanza de'
Parti, corse a cavallo a Vicenza con un solo scudiere. Giunto in città,
dopo avere due volte mutato cavallo, chiamò i suoi compagni d'armi;
e non fermandosi che il tempo necessario per bevere un bicchiere di
vino che gli fu presentato da una povera femmina, fece aprire la
porta di Liseria e piombò sui Padovani con soli cento uomini d'armi
ch'eransi adunati intorno a lui. Tutta l'armata padovana era occupata
nel saccheggio o nella dissolutezza. Cane non trovò nel sobborgo
veruna resistenza; alquanto più in là venne fermato un istante da un
piccolo corpo di gentiluomini, fra i quali trovavasi Albertino Mussato,
ma questo pure fu sgominato, ed Albertino scavalcato, fu fatto
prigioniere. A non molta distanza toccò la medesima sorte a
Giacomo di Carrara. Tutto il rimanente dell'armata più non pensò a
difendersi, ed era così grande il terrore de' Padovani, che Cane
trovossi, inseguendoli, con soli quaranta cavalieri, preso in mezzo da
cinquecento cavalli fuggitivi ch'egli si era lasciati addietro. Questi
ultimi sembravano agli occhi de' primi fuggitivi far parte dell'armata
di Cane, ed accrescevano il terrore; essi medesimi conoscevansi
posti tra due corpi nemici, e non osavano di far fronte. In questa
disfatta Vanne Scornazano, che l'aveva procurata, Giacomo e
Marsiglio di Carrara ed altri venticinque cavalieri con circa settecento
plebei furono fatti prigionieri. Il numero de' morti indica il
cominciamento di quelle guerre incruenti che avvilirono il coraggio

delle truppe italiane: non si trovarono sul campo di battaglia che sei
gentiluomini e trenta plebei
[331].
Dopo tale disfatta i Padovani cercarono di fortificarsi, chiamando in
loro soccorso gli alleati di Treviso, Bologna e Ferrara. Dal canto suo
Cane della Scala fece domandare rinforzi al capo del partito
ghibellino, ai Buonacorsi di Mantova, al duca di Carinzia ed a
Guglielmo da Castrobarco, coi quali credeva di potersi rendere
padrone di Padova. L'eccessive piogge, che inondarono tutta la
campagna, ritardarono dieci giorni tutte le operazioni militari.
Frattanto Cane della Scala riceveva alla sua corte i suoi più distinti
prigionieri, Giacomo di Carrara, Vanne Scornazano ed Albertino
Mussato. L'ultimo era nato nella più bassa classe del popolo, da cui
l'avevano innalzato i suoi talenti e la sua erudizione; ed era
risguardato come uno de' più letterati uomini del suo secolo.
«Peraltro, dice Ferreto di Vicenza, non era stato ancora decorato di
una corona di lauro o di ellera col titolo di poeta, non aveva ancora
pubblicata la sua storia, e la sua tragedia d'Ezelino non comparve
che dopo che gli fu dato il titolo di poeta. Ma egli amministrava già
con somma vigilanza gli affari della sua repubblica, ed in pari tempo
compilava con somma cura la storia de' fatti d'Enrico VII e de' mali
d'Italia. Era un uomo di vasti talenti, dotato di prudenza e di
facondia: non andò debitore che a sè medesimo, che ai proprj talenti
del titolo e della corona di poeta; perciocchè non essendo nato
d'illustri parenti non aveva ereditate nè ricchezze, nè credito nella
sua patria; ma sebbene uscito dall'ultima classe, fu dai tribuni e dai
magistrati del popolo innalzato al grado de' padri consolari ed ai
primi onori della repubblica Padovana. Egli ricevette per compenso
de' suoi talenti e delle sue fatiche grandissima fama e grandi
ricchezze, che gli furono assegnate sul tesoro pubblico
[332].» Per tal
modo il titolo di poeta, ed una capacità che oggi non ci sembra
singolare ottenevano allora non solo la gloria, ma ancora le ricchezze
ed il potere. Al presente le poesie del Mussato e la sua tragedia non
lo salverebbero dall'obblio; la sua stessa storia è riputatissima solo
per essere contemporanea, e malgrado la molta luce che sparge

intorno ai più importanti avvenimenti di quei tempi, il nome del
Mussato non è noto che a pochi eruditi.
Frattanto la sospensione delle ostilità che non era che una
conseguenza delle inondazioni, e le frequenti conferenze dei capi de'
Padovani con Cane della Scala, ridussero le due parti a proposizioni
di pace. Allora fu che Giacomo da Carrara contrasse segreta amicizia
con Cane, onde fu posto in libertà per trattare personalmente
intorno alla pace nella sua patria.
Giacomo di Carrara ammesso nel senato di Padova dovette disputare
contro Macaruffo, capo de' patriotti, che diffidava della sua
ambizione. Non voleva Macaruffo che la repubblica compromettesse
l'onor suo accettando la pace dopo una disfatta; ma erano così eque
le proposizioni di Cane, che non erano ingiuriose a Padova: ogni città
doveva tornare in possesso del suo antico territorio; i diritti
patrimoniali dei cittadini padovani nel distretto di Vicenza dovevano
essere loro restituiti; e la repubblica di Venezia veniva chiamata
garante del proposto trattato. A tali onorevoli condizioni la pace fu
infatti accettata dal senato di Padova, e sottoscritta il 20 ottobre del
1314
[333].
Questa pace per altro non ebbe lunga durata: i Padovani cercavano
opportunità di vendicarsi dell'avuta disfatta; i Vicentini soffrivano
impazientemente il giogo di Cane della Scala e domandavano spesso
ai loro vicini di ajutarli a scuoterlo. Macaruffo ed i suoi partigiani
favorivano i Vicentini malcontenti; ma Giacomo da Carrara era
segretamente attaccato a Cane. I primi si fecero lecito di entrare
senza il consentimento della repubblica in una congiura, che doveva
esserle cagione di grandi calamità.
Il 21 maggio del 1317 gli esiliati di Vicenza, quelli di Verona e di
Mantova ed i loro partigiani di Padova, che avevano prese le armi per
soccorrerli, si portarono di notte presso ad una porta di Vicenza che
alcuni traditori avevano promesso di consegnar loro: ma essi
medesimi erano traditi da coloro che credevano aver guadagnati col
danaro. Cane, avvisato del loro arrivo, gli stava aspettando in città; e
quando duecento di loro ebbero passato la porta, piombò sopra di

loro e tutti gli uccise o fece prigionieri. In seguito attaccò gli altri
rimasti al di fuori, li ruppe, e gl'incalzò fino sul territorio di
Padova
[334].
Cane della Scala si lagnò d'avere i Padovani rotta la pace con lui
conchiusa, e domandò che la repubblica di Venezia gli obbligasse a
pagare venti mila marchi d'argento; pena imposta a coloro che
commettessero le prime ostilità. Dal canto loro i Padovani
assicuravano di non aver presa parte nella congiura che non era
stata diretta che dai fuorusciti; ma Cane, dopo avere condannati a
morte cinquantadue congiurati fatti da lui prigionieri, venne colla sua
armata a guastare il territorio di Padova; e prima che terminasse la
campagna s'impadronì dei forti di Monselice, di Montagnana e di
Este
[335]. Anche nell'inverno e nella susseguente primavera continuò
a guastare le campagne de' Padovani, senza che questi fossero a
portata di fargli resistenza. Risparmiò per altro le terre appartenenti
alla casa da Carrara; ma era tale la leggerezza del popolo padovano,
che a quest'epoca aveva collocata tutta la sua confidenza nella
medesima casa da Carrara; e rimproverando Macaruffo d'avere
eccitata una così disastrosa guerra, lo sforzò a cercare, con tutti i
veri patriotti, sicurezza nell'esilio. Finalmente come la repubblica
soffriva ogni giorno nuovi mali, i partigiani dei Carraresi, che
occupavano soli tutte le magistrature, adunarono il senato dei
decurioni, onde provvedere ai pericoli della patria. Poichè molti
senatori ebbero parlato delle tristi circostanze in cui trovavasi lo
stato, Rolando di Placiola giureperito si levò: «Qual bisogno di più
lungo discorso, diss'egli, o cittadini! il rimedio per noi salutare e per
la nostra patria è bastantemente conosciuto. L'abuso de' plebisciti
l'abbiamo provato, egli ci conduce a certa ruina; proviamo una volta
se le leggi di un solo uomo ci possono procurare miglior sorte. Ogni
cosa sulla terra è sottomessa ad una sola volontà; le membra
ubbidiscono alla testa; le mandre riconoscono un capo. Se tutto il
mondo dipendesse da un re giusto si vedrebbero cessare le
carnificine, la guerra, la rapina e tutte le vergognose azioni. Siamo
docili alle voci della natura, seguiamo l'esempio che ci dà; facciamo
tra noi scelta del nostro principe. Egli solo si prenda cura del

governo, moderi la repubblica colla sua volontà, stabilisca le leggi,
rinnovi gli editti, abolisca quelli che più non si osservano; egli sia, in
una parola, il signore, il protettore di tutto quanto ci
appartiene
[336].» Con questi luoghi comuni un partigiano del
despotismo determinò il popolo, stanco di tante agitazioni, a privarsi
della propria esistenza. Il suicidio politico si compì; niuno rispose al
discorso del Placiola, e Giacomo da Carrara fu universalmente
indicato come il solo capace di comandare alla nazione. Non si
contarono i suffragi, secondo l'antica costumanza, con palle segrete;
ma con una acclamazione, che parve universale, Giacomo da Carrara
fu proclamato principe di Padova. Circondato dai consiglieri,
presentossi egli al popolo sulla piazza pubblica, ove Rolando della
Placiola replicò il suo discorso; e le acclamazioni de' partigiani della
casa di Carrara, che occupavano tutte le uscite della piazza, parvero
approvare la risoluzione presa dal senato. Così ebbe fine la
repubblica di Padova, e cominciò il principato dei Carraresi il 28 luglio
del 1318
[337].
Non abbiamo annoverata tra le libere città dell'Italia settentrionale
quella di Cremona, sebbene di que' tempi si governasse a comune;
ma questa città, lacerata da interne fazioni, aveva così
frequentemente mutato governo e tante volte era venuta in dominio
d'un solo, che non conosceva la libertà più di quello che la
conoscessero le città da lungo tempo cadute in servitù. Quasi nello
stesso tempo di Padova, Cremona rinunciò di nuovo e solennemente
al governo popolare.
Cremona era stata ruinata dall'imperatore Enrico VII e non erasi più
rialzata dal colpo allora ricevuto: il territorio di questa città era
affatto aperto, atterrate le fortificazioni de' suoi castelli e villaggi; e
nella crudel guerra ch'eransi fatte in quest'epoca le nemiche fazioni,
aveva la città medesima perdute in gran parte le sue ricchezze e la
sua popolazione. Cane della Scala, signore di Verona, e Passerino dei
Bonacorsi, signore di Mantova e di Modena, progettarono di
sottomettere questa città e quelle di Parma e di Reggio. Erano tutte
tre governate dal partito guelfo e sembravano situate a posta loro.
Convennero di dividerle tra di loro, ed attaccarono prima delle altre

Cremona, siccome la più debole e la più vicina
[338]. Durante l'estate
del 1315, guastarono il territorio cremonese, occuparono molti
villaggi che non poterono resistere, altri ne presero d'assalto,
trucidandone gli abitanti. I Cremonesi tormentati dalla fame e dalla
miseria, col nemico alle porte, perciocchè Cane si era innoltrato fino
al sobborgo di Cossa, e col territorio tutto guasto, tranne pochissimi
villaggi, erano inoltre agitati da intestine discordie. Il popolo
attribuiva ai grandi le sventure della repubblica ed andava dicendo
che per mettere fine alle loro dissensioni conveniva dare un capo allo
stato; che per difendere i popoli dall'attuale maniera di trattare la
guerra, non era vi che il governo d'un solo; che Verona, Parma,
Mantova, Milano, quasi tutte le città della Lombardia, offrivano un
esempio ch'era omai tempo d'imitare; che tornerebbe minore
vergogna ai Cremonesi dall'ubbidire ad un loro concittadino, che a
Cane o a Passerino; e che un principe potrebbe solo far cessare gli
odi che avevano fatto spargere tanto sangue e mandare in esilio
tanti cittadini.
Frattanto il partito repubblicano cercava di protrarre l'esecuzione di
così funesto consiglio; ed alla testa degli amici della libertà Ponzino
Ponzoni, capo dei Ghibellini, andava ripetendo che preferiva di
vedere la sua patria preda delle fiamme, piuttosto che sotto il giogo
di un tiranno
[339]. Malgrado la sua resistenza scoppiò tra la plebe
una sedizione il 5 settembre del 1315. Giacomo marchese Cavalcabò
fu condotto al pretorio dai sediziosi e proclamato signore della città.
Gli amici della libertà si ritirarono ne' villaggi e gli eccitarono alla
sommossa: Ponzino Ponzoni, citato da Cavalcabò a tornare in città,
rispose; «non aver fin allora combattuto contro i nemici dello stato
che per sottrarsi alla servitù; e non sapere adesso quale motivo
potrebbe mettergli le armi in mano contro gli stranieri, mentre la
scure della tirannide stava sospesa sopra tutte le teste; che per
ultimo non riconosceva altra patria che Cremona libera.»
L'opposizione del Ponzoni a questa infelice risoluzione non tardò ad
essere giustificata dagli avvenimenti; dopo sei mesi le guerre civili
forzarono il marchese Cavalcabò a rinunciare la signoria tra le mani
di Giberto da Correggio; le guerre esterne colmarono la miseria di

Cremona; ed il giorno 17 gennajo del 1322, impadronitosene
Galeazzo Visconti, la riunì alla signoria di Milano
[340].
Molte delle città della Lombardia e della Marca erano governate dai
signori, senza per altro avere rinunciato ad ogni desiderio di libertà.
Tante violenze erano state commesse in nome dei due partiti guelfo
e ghibellino, accesi tanti odj, tante vendette provocate, che il primo
desiderio dei cittadini e specialmente dei gentiluomini, era il trionfo
della propria fazione e la proscrizione degli avversarj. Una selvaggia
indipendenza era per loro preferibile alla libertà; essi misuravano i
loro diritti colle loro forze, e non supponevano che potessero essere
limitati dalle leggi. Nelle città poste nel centro della Lombardia, in
mezzo a quelle vaste campagne che avevano dato tanto vantaggio
alla cavalleria dei gentiluomini sopra l'infanteria de' borghesi, in
Cremona, Crema, Lodi, Piacenza, Pavia, Parma, Modena e Reggio,
non eravi durevole tirannide assicurata ad una sola casa, perchè
l'eguaglianza delle forze dei due partiti guelfo e ghibellino, non
lasciava a veruna usurpazione il tempo di consolidarsi; ma non
perciò eravi maggior libertà che altrove. Ogni anno veniva
contraddistinto da qualche nuova rivoluzione; per altro soltanto
cambiavansi gli uomini senza che il governo lasciasse mai d'essere
militare e dispotico. A popoli divisi in partiti che mai non posavano le
armi, erano necessari capi assoluti, e quand'ancora proclamavansi
talora i nomi di libertà e di repubblica, e ripetevansi per le contrade il
grido di popolo, popolo, per iscacciare un tiranno diventato esoso ai
cittadini, non per ciò si ristabiliva un libero governo. I consigli non
erano organizzati con abbastanza di forza perchè potessero
ricuperare la sovranità, non conoscevasi omai che l'autorità
degl'individui, e gli atti arbitrari non venivano più risguardati dai
cittadini quale violazione dell'ordine sociale; non credevano illegale
tutto quanto non era ingiusto; ed applaudivano sempre ai podestà
ed ai giudici che castigavano i colpevoli, quand'ancora
l'amministrazione della giustizia era nelle loro mani diventata
arbitraria, e che disprezzavano tutte le forme prescritte dalle leggi
andate in dissuetudine.

Per altro allorchè qualche vittoria faceva entrare un capo di parte in
una di queste città, sebbene i suoi partigiani lo rivestissero del
potere militare e delle attribuzioni giudiziarie de' podestà, non però
doveva trovare abbastanza soddisfatta la sua ambizione: i suoi
partigiani volevano essere troppo indipendenti; i suoi nemici,
quantunque esiliati, non cessavano di essere pericolosi, tenendosi
sempre armati; l'esempio de' suoi predecessori e de' vicini lo
avvertiva che l'autorità sovrana era di breve durata, e che, lungi dal
poterla trasmettere ai suoi figliuoli, non potrebbe conservarla egli
medesimo fino alla morte. Tale incerta situazione eccitava tutte le
passioni di un ambizioso, il quale, dopo essersi innalzato coi talenti
militari, cercava di assicurarsi l'usurpata autorità con una politica, ora
perfida, ora crudele. Il marchese Cavalcabò a Cremona, Alberto
Scotto a Piacenza, Venturino Benzone a Crema, Giberto da Correggio
a Parma, Matteo Visconti a Milano, Manfredi Beccaria e Filippone di
Langusco a Pavia, ed altri venti tiranni occupavansi sempre di così
fatte trame. Abbiamo abbandonate le particolarità degli oscuri loro
complotti, che altro non sono che una lunga serie di tradimenti. Le
frequenti ripetizioni degli stessi atti di slealtà avevano accostumati i
tiranni a non vergognarsene, i popoli a non maravigliarsene: l'arte di
tradire riputavasi abilità, e la crudeltà un utile mezzo d'ispirar timore.
Pure non è che in mezzo ad una società virtuosa che il delitto può
condurre con maggior sicurezza al principato; perciocchè quando
tutti disprezzano egualmente la morale, il tradimento punisce il
tradimento; il delinquente riclama invano a favore del nuovo suo
stato la guarenzia sociale ch'egli stesso ha distrutta; ogni colpevole
può rimproverarsi d'avere gratuitamente violate le leggi protettrici di
tutti; e la perdita del sentimento e della venerazione della giustizia
trae seco la perdita per tutto il popolo d'ogni prosperità.
Le città del centro della Lombardia erano in allora, non v'ha dubbio,
le più infelici dell'Italia: governate con una mano di ferro da signori
di breve durata che ispirare non potevano che orrore o disprezzo,
vedevano continuamente il loro territorio in preda alla guerra civile:
molte castella mantenevansi sempre ribelli contro la capitale; gli
emigrati che vi si rifugiavano, uscivano frequentemente per guastare

le campagne ed abbruciare le messi, e si trovava più facile il punire
questi saccheggi colle rappresaglie, che non il reprimerli col mezzo
delle armi. Non conoscevasi l'esempio di verun signore che avesse
potuto conservare più di dieci anni la signoria d'una città; ed ogni
rivoluzione, preceduta da una zuffa che costava la vita a molti
cittadini, era accompagnata dall'esilio e dalla ruina di tutto un
partito, di cui venivano confiscati i beni e spianate le case.
Non pertanto in mezzo a tanti disastri la popolazione non diminuiva
sensibilmente, nè spegnevasi affatto l'energia nazionale. Eravi troppa
vita in tutte queste zuffe, troppe passioni in movimento perchè ogni
individuo non sentisse il bisogno di sviluppare tutto il suo essere, di
fidarsi alle forze proprie, piuttosto che a quelle della società, e di
conservare la sua morale indipendenza sotto la servitù politica.
L'avvenire che sotto un despotismo stabilito non presenta veruna
mutazione ad un padre di famiglia, ne offriva mille tra le rivoluzioni
di questi tiranni di un giorno. Tutti i cittadini invidiavano non solo la
sorte di quelle repubbliche in cui la costituzione guarentiva la
sicurezza colla libertà, ma perfino la sorte degli stabili principati, ne'
quali almeno godevasi il riposo; ma per altro restava loro almeno la
speranza, mentre non vi è più speranza sotto un despotismo
costituito.
Contavansi di già alcune città ove qualche famiglia aveva stabile
signoria, e dove l'ereditaria successione di due o tre generazioni
pareva averne legittimato il dominio. La casa d'Este regnava a
Ferrara dall'epoca dello scacciamento dei Salinguerra e della disfatta
dei Ghibellini, accaduta del 1240, fino alla morte d'Azzo X nel
1308
[341]. A quest'epoca venne spogliata della sua sovranità dai
Veneziani e dal papa, che da prima avevano in qualità d'ausiliari
preso parte in una disputa di successione. Frattanto i marchesi
d'Este furono richiamati del 1317 alla sovranità di Ferrara
dall'attaccamento del popolo. Una casa meno illustre, quella de'
Bonacorsi, erasi impadronita nel 1275 della sovranità di Mantova, e
dopo averla conservata cinquantatre anni, cedette il posto ai
Gonzaga, che seppero mantenersene signori più lungo tempo
d'assai. Martino della Scala erasi innalzato in Verona al supremo

potere, del 1260, sopra le ruine della casa da Romano, e sebbene
del 1277 fosse stato ucciso dai congiurati, la sovranità come una
eredità legittima passò a suo fratello, indi ai figli del fratello. L'anno
1275 Guido Novello da Polenta era stato dichiarato signore di
Ravenna, che senza nuove rivoluzioni restò in potere della sua
famiglia. Finalmente la casa da Camino succedeva a Treviso, Feltre e
Belluno alla famiglia d'Ezelino di cui era stata sì lungo tempo rivale.
Eranvi dunque in Italia alcuni esempi d'una monarchia ereditaria
riconosciuta dai popoli e che conservavasi piuttosto col loro tacito
consenso che colla forza.
Ma queste dinastie, in allora risguardate come antiche in confronto
delle altre, erano ancora nuove paragonate all'ordinaria durata
degl'imperj. Le più non erano giunte alla terza generazione; il
principe non poteva dispensarsi d'essere soldato, veniva educato in
mezzo alle armi ed era forzato di governare egli stesso sotto pericolo
d'essere balzato dal trono dal favorito cui si fosse confidato. La casa
d'Este non venne spogliata de' suoi stati che per essere, siccome più
antica delle altre, la più corrotta di tutte. Soltanto cinquant'anni dopo
noi vedremo regnare que' tiranni voluttuosi, deboli, pusillanimi,
indegni successori de' guerrieri fondatori delle loro dinastie.
Taluno di questi piccoli principi accordò ben tosto la sua protezione
ai letterati. Fino nel precedente secolo i marchesi d'Este avevano
chiamato alla loro corte i trovatori ed i poeti provenzali. Dante in
tempo del suo esiglio trovò asilo e protezione presso molti signori
della Lombardia: a Ravenna Guido da Polenta, il marchese Malaspina
in Lunigiana, e più d'ogni altro i signori della Scala, in Verona lo
accolsero cortesemente. Can grande, che vedremo in appresso
sollevare questa casa ad un altissimo grado di potenza, manifestò in
principio del suo regno il suo amore per le lettere ed aprì nella sua
corte un onorato ricovero a tutti i fuorusciti illustri d'Italia. Uno di
costoro accolti da Can grande era lo storico di Reggio, Sagacio Muzio
Gazzata, che ci tramandò la relazione del trattamento che i dotti
avevano nella corte di Cane
[342]. «Diversi appartamenti venivano
loro, secondo la diversa loro condizione, assegnati nel palazzo del
signore della Scala, e tutti avevano domestici e mensa

elegantemente imbandita. I varj appartamenti erano indicati da
simboli e da insegne; il trionfo pei guerrieri, la speranza per gli esuli,
le Muse per i poeti, Mercurio per gli artisti, il paradiso per i
predicatori. In tempo del pranzo, musici, buffoni, giocolieri, giravano
in questi appartamenti; le sale erano ornate di quadri rappresentanti
le vicende della fortuna, e Cane talvolta invitava alla propria mensa
alcuno de' suoi ospiti, specialmente Guido di Castel di Reggio, che
per la sua semplicità chiamavasi il semplice lombardo
[343], ed il
poeta Dante Alighieri.» Senza dubbio tra i proscritti guerrieri
eranvene pochi cui la camera de' trionfi appartenesse a più giusto
titolo che ad Uguccione della Fagiuola, cui Cane diede asilo dopo che
questo capo di parte perdette la sovranità di Lucca e di Pisa. Colà
Dante legò con costui strettissima dimestichezza, e prese occasione
di dedicargli la prima parte del suo poema
[344].
La protezione che con tanta frequenza i principi accordano ai poeti
piccoli sacrificj loro costa e procaccia loro molta celebrità. In ogni
tempo, in tutti i paesi, i poeti misurarono la loro ammirazione per un
principe sulle sue liberalità; e non arrossirono di rendere coi versi
immortali le vili loro adulazioni, come non ebbero vergogna di
riceverne il salario. Non dobbiamo perciò essere sorpresi, se in
questo e nel susseguente secolo i più distinti poeti italiani
frequentarono la corte de' principi, dai quali erano festeggiati assai e
più splendidamente pagati che dalle repubbliche. Ma per altro i poeti
non hanno potuto sorgere che ne' tempi in cui lo spirito di libertà
animava alcuna delle parti della sacra terra d'Italia, che durante il
tempo che nella stessa lingua altri trattavano le quistioni che
decidono della prosperità e della gloria degli uomini. Quando la via
del pensiero fu chiusa agl'Italiani, si spense ancora la loro
immaginazione. Un padrone non può scegliere tra le facoltà dello
spirito umano, non può dire a' suoi sudditi: abbiate immaginazione e
non intendimento; io vi concedo la poesia, ma vi rifiuto la filosofia; vi
permetto la fisica e vi proibisco la morale; vi lascio le scienze esatte,
ma prendete cura di non toccare la politica. È necessario di togliere
lo steccato che inceppa lo spirito umano, o rassegnarsi alla sua
indolenza, alla sua apatia. Dopo perduta la libertà, una sola

generazione può ancora agitarsi per cercare l'apparenza della gloria
in quegli esercizj dello spirito che un despota gli permette ancora;
una seconda generazione dopo la caduta di questa può ancora
distinguersi nelle belle arti che conservano un simbolo del pensiere,
senza esprimerlo in un modo formidabile pel tiranno; ma gli avanzi di
questa sacra fiamma non possono in verun modo conservarsi un
intero secolo dopo spenta la libertà; è tolto loro lo scopo delle
umane generazioni, sono mancati i motivi de' loro sforzi: non avvi
più gloria quando viene dispensata dal favore d'un principe e divisa
tra i suoi servitori ed i suoi poeti.
Gli artisti più festeggiati dai principi ereditari che si credettero al
sicuro da ogni rivoluzione, furono gli architetti. I marchesi d'Este, gli
Scaligeri, i Visconti, cominciarono assai presto ad innalzare que' vasti
e sontuosi edifici che attaccano tuttavia qualche gloria alla loro
memoria, sebbene la ricordanza delle loro azioni sia quasi affatto
spenta. Le città libere avevano adottato il lusso dell'architettura; per
lo contrario i violenti usurpatori non lasciarono che ruine, avendo
avuto bisogno di tutte le loro forze, delle loro ricchezze, pel
momento presente, onde non osarono di pensare all'avvenire. Nella
seconda generazione i signori ripigliarono il gusto dell'architettura,
che diventò pure in loro mano un oggetto di politica, credendo di
dovere far pompa della propria grandezza per farsi rispettare dai loro
sudditi ed ispirar timore ai nemici. Avevano bisogno di un'idea di
perpetuità per assodare il loro dominio, e perchè loro non bastava il
tempo passato, prendevano possesso de' vegnenti secoli con edifici
destinati all'eternità.
Il lusso di queste piccole corti, le spese che facevano i re d'una città,
per la loro guardia, per l'armata, per gli edifici, pei regali che davano
ai buffoni ed ai cortigiani, provano l'ammasso di grandi ricchezze.
Vero è che la maggior parte de' signori erano stati ricchissimi
proprietari avanti che diventassero padroni della loro patria; e che
aggiugnevano l'entrate dell'antico patrimonio ai pubblici tributi
stabiliti ne' tempi della libertà; imperciocchè sembra che non
osassero di accrescerli, non essendo mai giunti ad ottenere il credito
di cui godevano le città libere, sicchè potessero supplire col prestiti ai

bisogni improvvisi dello stato. Un'imposta territoriale, descritta in
ogni signoria sopra un catastro, formava parte di quest'entrata,
un'altra più importante parte era pagata dagli abitanti delle città in
forza di una gabella posta sulle derrate che vi si consumavano e per
un diritto d'entrata ed uscita delle mercanzie provenienti dall'estero,
o mandate all'estero; poichè il prodotto dell'industria del paese non
era esente dalle tasse. Del rimanente non erasi ancora inventato
verun sistema di protezione per il commercio e per le manifatture;
onde in mezzo alle guerre ed alle rivoluzioni, il commercio e le
manifatture prosperavano infinitamente meglio che non al presente
in que' canali artificiali, in cui le moderne nazioni vollero forzarli ad
entrare. Tutte le città lombarde fabbricavano drappi di lana; i quali,
oltre all'interno consumo, bastavano ad una ragguardevole
esportazione che facevasi per mezzo de' Veneziani
[345]. Le
manifatture di lana erano state fondate in Lombardia dai monaci
umiliati. A Milano il convento di Brera, diventato oggi il palazzo delle
scienze e delle lettere, era la grande officina della fabbrica dei
drappi, ed i monaci di questo convento l'anno 1309 si obbligarono,
per una somma di danaro, a mandare una colonia per istabilire
un'eguale manifattura in Sicilia, mentre i Milanesi apprendevano dai
Siciliani l'arte di lavorare la seta
[346].
I sudditi de' principi di Lombardia oramai si limitavano alle sole
manifatture. Dopo la perdita della libertà essi più non si recavano in
Francia, nelle Fiandre, in Inghilterra, come solevano fare ancora i
Veneziani ed i Toscani; non aprivano più banco in ogni città, non
s'impadronivano più del commercio di banco e di quello dei trasporti
dell'Occidente. Il nome di Lombardi, che i Francesi, invidiando tanta
attività, avevano dato ai prestatori sopra pegno non era più
meritato: i soli Fiorentini e Lucchesi, non già gli abitanti di Asti, di
Milano e d'Alessandria, esercitavano, come per lo passato, questo
mestiere. Abbiamo già dovuto avvertirlo, parlando della Grecia, che il
commercio straniero che domanda lunghi viaggi e vaste
combinazioni, non può intraprendersi e sostenersi senza una certa
energia di carattere, senza uno sforzo dello spirito, che non si

trovano nella mezzana classe d'una nazione, fuorchè presso un
popolo libero.
Del rimanente, in questi piccoli principati, il popolo vivea piuttosto
rassegnato che contento, più non si occupando della sua futura
sorte, nè di timori, nè di speranze. Rientrato in quella oscurità da cui
l'avevano fatto uscire le precedenti agitazioni, non lasciava dietro di
sè veruna orma, verun nome che si sollevasse al di sopra degli altri;
e la storia nelle città sottomesse alle nuove dinastie, più non può
risguardare che una sola famiglia, e spesse volte un solo individuo.
FINE DEL TOMO IV.

TAVOLA CRONOLOGICA DEL TOMO IV.
Capitolo XXIII. Guerra di Sicilia. — Grandezza e
decadenza della repubblica di Pisa. — Crudel
morte del conte Ugolino. — Nuove turbolenze
a Firenze. 1282-1292. pag. 3

Anno

Carlo d'Angiò non doveva, a quanto
pare, essere troppo indebolito dai
vesperi siciliani ivi

Mezzi di resistenza che una
passione nazionale dà ai Siciliani 4

Gli abitanti di Palermo tentano di
placare il re ed il papa 6
1282Il 6 luglio. Carlo attacca Messina
con una flotta ed una armata
imponenti 7

Il 30 agosto. Pietro d'Arragona
giugne a Trapani, e riceve
omaggio dai Siciliani 8

Ruggeri di Loria, ammiraglio de'
Siciliani, occupa lo stretto di
Messina 9

Diffida vicendevole de' re
d'Arragona e di Napoli 10
1282Carlo costretto ad abbandonare la
Sicilia ed a tornare in Calabria 12

La sua flotta viene incendiata alla
Catona e a Reggio da Ruggeri di
Loria ivi

Carlo propone a Pietro un
combattimento in campo chiuso13

Gli apparecchi per tale
combattimento lasciano per
qualche tempo la Sicilia in riposo14
1276-1282I Pisani in tempo di pace acquistano
maggiori ricchezze e potenza ivi

Rivalità de' Pisani e de' Genovesi;
dispute tra queste popolazioni in
Corsica 16
1282Le flotte dei due popoli si
minacciano qualche tempo senza
battersi 17

Disastro della flotta di Ginicello
Sismondi 18

Esploratori mantenuti
pubblicamente dalle due città 19
1283Potenti flotte dei Pisani e dei
Genovesi che si minacciano senza
venire a battaglia 20
1284Il 1 di maggio. Guido Jacia,
ammiraglio pisano, battuto da
Enrico de' Mari 21

I Pisani armano a spese de'
particolari una flotta di centotre
galere 22

Il 6 agosto. Battaglia della Meloria
tra i Genovesi ed i Pisani 24
1284Accanimento di questa battaglia.
Oberto Doria, ammiraglio
25

genovese, si batte con Alberto
Morosini, ammiraglio pisano

Rotta de' Pisani colla perdita di
cinquemila morti ed undicimila
prigionieri 26

Costernazione generale de' Pisani
quand'ebbero notizia della
disfatta 27

I Genovesi ricusano di ricevere la
taglia per la libertà dei Pisani, che
tengono 16 anni prigionieri 28

Il 10 novembre. Lega de' Guelfi
toscani per attaccare Pisa 29
1285Il conte Ugolino della Gherardesca
nominato capitano generale di
Pisa 31

Egli riesce a disciogliere la lega de'
Guelfi toscani contro Pisa ivi

Cerca di liberare i prigionieri,
cedendo Castro di Sardegna, ma
s'oppongono gli stessi prigionieriivi

Ottiene la pace dai Lucchesi
cedendo loro molti castelli 32

Il conte Ugolino prende a
perseguitare i Ghibellini 33

Nino di Gallura si associa ai suoi
nemici e cerca di muovere il
popolo contro di lui 34
1285-1287Il conte Ugolino rassoda la sua
tirannide 36
1285-1287Si riconcilia coi Ghibellini, e scaccia
dalla città Nino di Gallura 37

1288Violenza de' suoi trasporti di collera:
uccide un nipote dell'arcivescovo
Ruggeri 38

Il 1 di luglio. L'arcivescovo Ruggeri
l'attacca coll'ajuto de' Ghibellini 39

Il conte Ugolino viene chiuso co'
suoi figliuoli nella torre della fame40
1283Apparecchi per la battaglia in
campo chiuso che doveva aver
luogo a Bordeaux il 15 maggio 44

Papa Martino IV si oppone a questa
battaglia, ed Edoardo re
d'Inghilterra non vuole accordare
guarenzia ai due monarchi ivi

Carlo si porta a Bordeaux; Pietro
protesta di non esservi per lui
sicurezza 45

Il 15 marzo. Sentenza del papa che
spoglia Pietro dei regni di Sicilia e
d'Arragona 47
1284Carlo torna per mare a Napoli 48

Il 5 maggio. Prima del suo arrivo a
Napoli, suo figlio Carlo vien fatto
prigioniere da Ruggeri di Loria49

Carlo d'Angiò punisce severamente i
Napoletani malcontenti 50
1284Si lascia ingannare dalle
negoziazioni de' Siciliani, e perde
la stagione propria ad agire 51
1285Cade infermo a Foggia e muore il 7
di gennajo in età di
sessantacinque anni ivi

Il 25 di marzo. Morte di Martino IV,
cui succede Onorio IV 52
1282Nuova costituzione dei Fiorentini; i
priori delle arti e della libertà 54

I priori ne' due mesi della loro
carica sono prigionieri in palazzo 56
1283Rivoluzione a Siena; stabilimento
della signoria e dell'ordine dei
nove 57

Eguale rivoluzione in Arezzo, seguita
nel 1287 da una controrivoluzione58
1288I Ghibellini di Pisa e d'Arezzo
dichiarano la guerra ai Guelfi ed
ai Fiorentini 60
1289L'undici giugno. Rotta degli Aretini a
Certomondo, presso di
Campaldino ivi
1289-1293Vantaggi ottenuti dai Pisani sotto la
condotta del conte Guido da
Montefeltro 61
1292Dissensioni in Firenze tra i nobili ed
il popolo 63

Giano della Bella, gentiluomo
fiorentino, capo del partito
popolare 64
1292Ordinanza di giustizia che rende
sottomessa la nobiltà 66

Organizzazione militare della città; il
gonfaloniere della giustizia 67

Dino Compagni, gonfaloniere,
atterra le case de' Galigai 69

Odio de' nobili contro Giano della
Bella; cercano il modo di perderloivi

Gli rendono nemici alcuni dei corpi
dei mestieri ivi
1294Accusano Giano della Bella avanti
ad una signoria di già resa loro
parziale 72

Il 3 di marzo. Giano viene esiliato, e
muore lontano dalla sua patria73

Capitolo XXIV. Pontificato di Bonifacio VIII. — Il
partito guelfo si divide in due fazioni, de'
Bianchi e de' Neri. — I Bianchi, perseguitati, si
uniscono ai Ghibellini. 1294-1303 75

1285-1287Pontificato d'Onorio IV ivi
1288-1292Pontificato di Nicola IV 76
1292-1294Vacanza della santa sede 77
1294Elezione di Pietro di Morone che
prende il nome di Celestino V 79

Celestino V fissa la sua residenza in
Napoli 81

Debolezza di questo papa e sua
assoluta incapacità di governare
la Chiesa 82
1294Intrighi di Benedetto Caietano,
cardinale d'Anagni, contro il papa83

Il 13 dicembre. Così consigliato dal
Caietano, Celestino si dimette
dalla dignità pontificia 84

Il 23 dicembre. Gli succede il
cardinale Caietano col nome di
Bonifacio VIII 85
1295Di gennajo. Pietro di Morone fugge
per tornare al suo eremitaggio 86

Bonifacio lo fa inseguire e chiudere
nella torre di Fumone 87
1296Il 19 maggio. Morte di Pietro di
Morone, ossia Celestino V 88
1294Il 10 dicembre. Tradizione intorno
alla Santa Casa trasportata a
Loreto 90
1291Il 19 di maggio. Melec Seraf
s'impadronisce di san Giovanni
d'Acri. Uccisione dei Cristiani92

Vani sforzi del papa per eccitare una
nuova crociata ivi
1288-1295Parzialità dei papi negli affari di
Napoli e di Sicilia 94

Carlo II, dopo uscito di prigione,
viene dal papa sciolto dal
giuramento che gli aveva
procurata la libertà 95

L'Arragona attaccata da Carlo di
Valois, la Castiglia e la Francia 96
1295Vergognoso trattato conchiuso colla
mediazione di Bonifacio tra
Giacomo re d'Arragona e Carlo II97
1296Protesta de' Siciliani contro il
trattato; essi nominano re
l'infante don Federico 98

Inutile tentativo di Bonifacio VIII
per negoziare con Federico 99

La guerra si rinnova con furore in
Calabria ed in Sicilia 100

Situazione di Pistoja. Carattere de'
suoi abitanti 101

Famiglie e fazioni a Pistoja de'
Cancellieri guelfi e de' Panciatichi
ghibellini 102

Tutti i nobili vengono esclusi l'anno
1285 dal governo di Pistoja 103

La famiglia de' Cancellieri si divide;
zuffa fra due membri della
medesima 104

Vendetta del ramo Nero de'
Cancellieri 105

Il ramo Bianco de' Cancellieri si
vendica a vicenda 106
1296-1300La città di Pistoja ed il suo territorio
si dividono tra i Cancellieri Bianchi
e Neri 107

Atti di crudeltà e di perfidia
commessi dalle due parti 108
1300La signoria di Pistoja ceduta per tre
anni ai Fiorentini quali pacificatori109
1300I capi delle due fazioni, Bianca e
Nera, vengono chiamati a Firenze110

Rivalità in Firenze tra Corso Donati
e Vieri de' Cerchi 112

I Donati si uniscono ai Neri di
Pistoja, i Cerchi ai Bianchi 113

Le due fazioni sempre
apparecchiate a venire alle mani114

Vieri de' Cerchi, il capo di parte
Bianca, manca di carattere 116

Bonifacio VIII cerca di metter pace
tra i due partiti 117

I capi de' Bianchi e de' Neri sono in
pari tempo esiliati da Firenze 118

Tornata de' Bianchi in Firenze;
intrighi de' Neri per vendicarsi 119

Il papa chiama in Italia Carlo di
Valois 120
1301I Bianchi opprimono il partito de'
Neri a Firenze ed a Pistoja 122

Il partito de' Neri trionfa a Lucca, e
fa esiliare Castruccio colla sua
famiglia 123

Carlo di Valois entra in Toscana per
le montagne di Pistoja 125

I Bianchi dispongonsi a difendersi a
Pistoja, ma non ardiscono di
attaccare Valois ivi

Questi va a Roma per concertare
ogni cosa col papa 126
1301Torna a Staggia e minaccia Firenze127

I Fiorentini acconsentono a riceverlo
sotto certe condizioni nella loro
città 128

Valois entra in Firenze
accompagnato da molta cavalleria129

Vieri de' Cerchi ed i Bianchi
trascurano i loro mezzi di difesa131

Valois non osserva le giurate
condizioni, e fa tornare gli esiliati
in Firenze 132

Fa imprigionare i Bianchi ed
abbandona le loro case al
saccheggio 133

Cante de' Gabrielli incaricato di
perseguitare il partito vinto ivi

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Dynamics Of Soft Matter Neutron Applications 1st Edition Alexei P Sokolov

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