Athene M Donald

1
Biological and Soft Matter Physics

• Athene M Donald
• University of Cambridge
• UK
• Experience
• Taught a 3rd year course, written from scratch,
  on Soft Matter and Biophysics for the last 3
  years.
• Previously taught a 4th year option on
  Structure and Properties of Condensed Matter for
  many years, which included much Soft Matter, as
  well as hard topics such as dislocations
• Chair of the Biological Physics Group, recent
  Committee member of the Polymer Physics Group,
  and member of the Liquids and Complex Fluids Group

2
Why Should we Teach these Subjects to our
Undergraduates?

• The EPSRC International Review of Physics
  (2005)said
• However, it is the perception of the Panel that
  there are quite a few physics departments where
  students get little, if any, exposure to modern
  soft matter physics. This is regrettable, because
  soft matter physics has deep links with many
  other areas of science, whilst the theoretical
  concepts and experimental techniques of this
  field are of direct relevance for biophysics. In
  addition, soft matter physics has many industrial
  applications.
• See also Ray Goldstein et al Physics Today March
  2005 Teaching Biological Physics
• Many physics departments teach a collection of
  undergraduate courses whose outlines are similar
  to the menu of 30 ears ago, despite the fact that
  the research interests of physics faculty have
  changed dramatically. The curriculum must change.
  In particular, we believe that biological physics
  must become a mainstream course in all physics
  departments, offered as regularly as, for
  examples, course on solid-state or high-energy
  physics.

3
What do we mean by Soft Condensed Matter?

• The term usually refers to states of matter which
  are neither simple liquids nor crystalline
  solids.
• Includes many familiar types of matter soap,
  yoghurt, paint, liquid crystals, putty.but also
  much of our bodies including cell membranes and
  the cytoplasm inside.
• In general we will be dealing with lengthscales
  intermediate between atomic and macroscopic
  these are often known as mesoscopic.
• Quantum mechanics will not therefore be useful
  the predominant techniques we will use will be
  statistical.
• Mean field theories will be found to be useful,
  as you have seen before, to describe the
  behaviour of large numbers of molecules.
• Although dealing with ensembles of molecules, we
  will find that thermal energy is comparable with
  the energies giving rise to distortion and
  interaction energies, so Brownian motion and
  fluctuations are very important.
• This is a key difference with 'hard condensed
  matter' for which thermal fluctuations are not
  important.

4
What do we mean by Biophysics?

• The term is used in different ways by different
  communities eg in Physiology you will find it
  referring to the study of electrical impulses
  across cell membranes.
• Many regard biophysics as a discipline still
  waiting to be adequately defined (Cotterill
  Biophysics an Introduction).
• Biophysics courses in Life Science degrees often
  cover topics where physics can be applied to
  biology, with the physics being kept simple.
• In this course we will be looking at where our
  detailed physics understanding, possibly gained
  from quite other systems, can be applied to study
  the complexity of biological systems.
• A key example is self-assembly.
• Biology (unlike physicists) is very good at this.

TMV Cell Membrane
5
Where does Soft Matter and Biological Physics fit
into the Syllabus?

• Soft Matter is not necessarily a natural
  companion to conventional Hard Condensed
  Matter.
• It more logically ties in with courses on
  Properties of Matter, Thermodynamics/Statistical
  Physics or Fluids (not just a Fluid Mechanics
  course).
• Where and how to place it depends on the other
  courses on offer, but parts of it can be put in
  in many places to introduce the concepts, without
  necessarily requiring a dedicated, early stage
  course.
• Students will be familiar with many examples of
  soft matter in everyday life ranging from
  foods, through adhesives, to liquid crystal
  displays.
• Every lecturer of any of the above sorts of
  courses ought to be able to fit in some soft
  matter consequently!
• Additionally, many students will be aware of the
  exciting developments in the life sciences and,
  even if committed to physics, be thirsty for a
  bit of knowledge from biology.
• Teaching biological physics can satisfy this yet
  keep them within physics.
• From the bums on seats viewpoint, it is in a
  departments interests to offer them the
  opportunity to learn about this material.

6

• Topics from the course I taught as a third year
  option no condensed matter (or fluids) had been
  taught by then, but plenty of thermodynamics.
• Soft Condensed Matter and Biophysics
• Liquids and Complex Fluids
• Brownian Motion
• Self Assembly
• Surface Energy
• Polymers and Biological Macromolecules
• Liquid Crystals

7
Biological Examples

• My course is not a true Biophysics course, but I
  believe it is very easy to put in a wide range of
  biological examples within a broad Soft Matter
  course.
• Other people may prefer to teach a course solely
  on Biological Physics (in Cambridge this is left
  to a 4th year option).
• It will obviously depend on how the whole
  syllabus is configured.
• The minimum I think is adequate is
• An introduction to the cell
• The cell membrane as an example of a
  self-assembled structure, which is easily treated
  as analogous to synthetic vesicles.
• The different key molecules in biology proteins,
  polysaccharides, nuclei acids and lipids.
• This naturally leads to ideas of self-assembly.
• This also means it is easy to tie in with ideas
  of nanotechnology, and I think there are many key
  exemplars one would give for that.

8
Starting from a Properties of Matter Course

• Gases
• Can use colloids as model hard sphere fluids
• Liquids
• Viscosity
• Viscoelasticity
• Reynolds number
• The difference for macroscopic and microscopic
  objects moving in fluids (nanobots, bacteria and
  rotating flagellae).
• The example of swimming through treacle.
• Solids
• Glasses, how to characterise amorphous materials
  via scattering
• Introduce different types of scattering, why
  neutron scattering is useful for polymers
• Protein crystals can be used as examples of
  crystals rather than simply copper and NaCl.
• Liquid Crystals sometimes described as the
  fourth state of matter.

9
Starting from a Statistical Physics Course

• The key concepts that need to be built on are
• Chemical potential
• Fluctuations
• And hence to the Fluctuation-Dissipation Theorem
• Because fluctuations are ubiquitous in soft
  matter, there are many examples that can be used
  to illustrate them.
• Membranes are a good example.

10
Starting from a Fluids CourseTopics that can be
brought in

• Biophysics
• Flow through pipes, Poiseuille flow and clogging
  of arteries.
• Stokes Einstein equation sedimentation studies
  for studying macromolecular size.
• Reynolds number and different strategies for
  propulsion at different sizes relevant to
  bacteria to sharks.

• Soft Matter
• Viscoelasticity of polymer Melts loss and
  storage modulus.
• Colloidal dispersions and shear thickening and
  shear thinning solutions.

Nanobot swimmers Golestanian et al, Sheffield
11
Useful Books

• RAL Jones Soft Condensed Matter OUP 2002
• D Tabor Gases, Liquids and Solids, 3rd ed CUP
  1991
• M Daoud and CE Williams, Soft Matter Physics,
  Springer 1999
• IW Hamley, Introduction to Soft Matter, Wiley
  2000
• P Nelson, Biological Physics, Freeman 2003
• KW Dill and S Bromberg, Molecular Driving Forces,
  Garland Science 2003
• Advanced Texts
• M Kleman and OD Lavrentovich Soft Matter Physics,
  an Introduction, Springer 2003
• PM Chaikin and TC Lubensky, Principles of
  Condensed Matter Physics, CUP 1995
• SA Safran, Statistical Thermodynamics of
  Surfaces, Interfaces and Membranes, Addison
  Wesley 1994
• M Daune, Molecular Biophysics, OUP 1999
• JN Israelachvili Intermolecular and Surface
  Forces, Academic 1985
• M Rubenstein and R Colby Polymer Physics,OUP
  2003
• PG de Gennes, F Brochard-Wyart and D Quéré,
  Capillarity and Wetting Phenomena Springer 2002

12
Additional Texts

• ME Cates and MR Evans, Soft and Fragile Matter,
  IoP 2000
• IM Ward Mechanical Properties of Solid Polymers,
  Wiley 1983
• IM Ward and J Sweeney, An Introduction to the
  Mechanical Properties of Solid Polymers, Wiley
  2004
• S Vogel Life in Moving Fluids Princeton 1994
• RAL Jones Soft Machines OUP 2004
• J Goodwin Colloids and Interfaces with
  Surfactants and Polymers Wiley 2004
• M Doi Introduction to Polymer Physics OUP 1992
• R Balescu Statistical Dynamics, Imperial College
  Press 1997
• D Boal Mechanics of the Cell, CUP 2002
• DC Bassett Principles of Polymer Morphology CUP
  1981
• AM Donald, AH Windle and S Hanna, Liquid
  Crystalline Polymers, CUP 2006
• JR Waldram, The theory of thermodynamics
• WCK Poon and D Andelman, eds, Soft Condensed
  Matter Physics in Molecular and Cell Biology,
  Taylor and Francis 2006

13
Conclusions

• Soft Matter and Biological Physics are both
  easily accessible topics which, even if a
  department chooses not to introduce as dedicated
  course(s) can be brought into many standard
  courses.
• Nevertheless, I firmly believe they should be
  offered as stand-alone courses because of the
  interesting physics they can teach.
• They can be made more or less theoretical to suit
  different students backgrounds.
• And they are topics many of the brightest
  students find very appealing.
• It has to be said in my own department,
  probably not coincidentally now we teach this
  material at third and fourth year levels, more
  students opt to do PhDs in these areas than in
  more conventional hard condensed matter areas
  (for which there are also courses). But this
  statement also applies to students from outside
  and abroad.

14
Colloids

• Colloids are systems in which one of the systems
  (at least) has dimensions of 1mm or less.
• Thus many aspects of nanotechnology are
  essentially colloidal.

• Examples
• Solid in
  liquid such as Indian Ink or sunscreen
• Suspension
• Liquid in
  Liquid such as mayonnaise or salad
  dressing
• Emulsion
• Gas in Liquid
  such as beer or soap foam
• Foam
• Gas in Solid such as
  bath sponge or ice cream
• Sponge

15
Packing and Excluded Volume

• Thus packing can be either regular or random,
  depending on circumstances.
• Phase transitions can be observed in colloids as
  a function of concentration, and different
  structures can coexist.
• Why should random packing sometimes be of higher
  free energy than crystals?
• The answer lies in the concept of excluded
  volume, and is similar to the argument for the
  existence of the hydrophobic force.
• To understand this consider a hard sphere
  colloid, analogous to a hard sphere gas.
• For an ideal gas
• where a is a constant.

• But if the atoms have finite volume b, the
  volume accessible is reduced to V-Nb
• Crudely
• at low volume fractions
• Thus per atom
• The finite size of the atoms gives rise to a
  repulsive term- the atoms cannot overlap.
• For colloids as well there is a similar
  effective excluded volume.
• The good packing in the crystal means there
  is more space for the atoms/colloids to explore
  thereby increasing entropy, despite the long
  range order.

16
Colloidal Crystals

• Sometimes you don't want the colloid to be
  stabilised!
• Well-ordered colloidal crystals can form, with
  the same symmetries as for atomic crystals.
• The optical properties of colloidal crystals form
  the basis for opals, in which aggregates of
  silica are dispersed in 5-10 of water.
• The local differences in packing give rise to
  optical effects giving precious opals their
  distinctive colours.
• Synthetic opals have much more regular packing.
• More generally they can be used as model systems,
  e.g as macroscopic hard sphere fluids to help
  physicists understand the nature of interactions.

• Polystyrene beads of diameter 700nm

17
Complex Viscosity and ViscoelasticityJones, Ward
Creep
A constant load is applied and the resulting
strain is measured.

• A viscoelastic material is, as the name
  suggests, one which shows a combination of
  viscous and elastic effects.
• Polymeric fluids and some solids are examples.
• The elastic term leads to energy storage.
• Its contribution to a shape change will be lost
  once the stress is removed.
• The viscous term leads to energy dissipation, and
  irreversible shape changes associated with the
  flow.
• Rate effects are very important for these
  materials.

• Load applied

0
Strain response
1
1
e1 immediate elastic deformation e2 delayed
elastic deformation e3 Newtonian flow (i.e.
permanent deformation Define creep compliance
18
Motion of E Coli cells

• E Coli cells examined by video-enhanced
  differential-interference contrast microscopy.
  Some cells are shown de-energised near the
  bottom of the preparation.
• They exhibit a variety of wave forms normal,
  colied , semic-coiled etc.
• The mobile cells show the strongly beating
  flagellae, which propel them forward.

19
Scattering as a Probe of Polymer Structure

• Scattered radiation may be electrons, X-rays,
  neutrons, light
• In all cases scattering arises due to contrast
  from fluctuations in concentration, electron
  densitydepending on the type of radiation used.

• The key parameter is the structure factor S(q),
  where q is the scattering vector (corresponding
  to scattering through some angle).
• I(q) is the intensity scattered through q, A is
  the scattering amplitude and N is the number of
  scatterers.
• The Fourier transform of S(q) is proportional to
  a correlation function.

20
Single Chain Scattering

• Define the structure factor by
• Now (Gaussian) average of exp ix
• And for our ideal Gaussian chain N repeats long,
  for each component i (x,y,z)
• So

• This tends to a double integral for large N
• f(x) is the Debye function, and Rg is the radius
  of gyration, a measure of the size of the polymer
• Then (for large N)

21
Experimental Results
SANS data PMMA MW 250000 in d-PMMA

• One can also use scattering to study polymer
  thermodynamics from looking at the scattering
  from concentration fluctuations.
• This provides a means for establishing c from the
  magnitude of the fluctuations (see Rubenstein and
  Colby for further details).
• This can be done for polymers in solution or for
  mixtures of polymers.

• o 0.31
• 0.63
• ? 0.93
• ? 1.19

Experimental data shows excellent agreement with
the Debye function (solid lines). This confirms
Gaussian statistics for long chains.
22
Liquid Crystals

• Crystals have 3 dimensional periodic structures.
• Amorphous materials including liquids are
  disordered.
• Liquid crystals have intermediate order, and are
  consequently sometimes known as mesophases.
• They have orientational but not positional order.
• They are neither true liquids nor crystals.
• There are various different types with differing
  symmetries.

• Friedelian Classes
• There are a range of different liquid crystals
  with different symmetries.
• Nematic lowest symmetry
• n is the preferred direction, known as the
  director 
• Alignment with the director is not perfect.

23
Smectics

• Smectic C
• n inclined to layer normal
•  
• There are a whole series of smectic phases, with
  different degrees of symmetry.
• Some of them are really equivalent to
  low-dimensional crystals.
• All have layer structures.

• Smectic from Greek for soap.
• Layer structure with nematic order within
  layers.
• Smectic A
• n parallel to layer normal

n
layer
24
Cholesterics

• Cholesteric the name comes from cholesterol.
• Chiral molecules (i.e. ones with asymmetric
  carbon atoms, that is the molecule differs from
  its mirror image) spontaneously twist.
• Nematic order in each layer, but there is a
  (systematic) angular twist between successive
  layers.
• This leads to a helical structure.
• Helix has a well-defined pitch.

Half turn of helix director has rotated by p.
25
Order Parameter and Order Parameter Tensor

• Idea most easily applied to nematics.
• Describes how good the alignment is with respect
  to the overall director.
• The order can be characterised by an order
  parameter.
• where qi is the angle the ith molecule makes with
  n.
• This is equivalent to the 2nd Legendre polynomial
  P2.
• It is a scalar quantity.
• S 1 for perfect alignment
• S 0 for random alignment (as in isotropic
  liquid) and

• However, in the presence of external fields which
  may not be aligned with the director, need a more
  formal analysis.
• Nematic phases either have an inversion centre,
  or equal probabilities of pointing up or down
  there are no ferroelectric nematics.
• If na is the unit vector pointing along the
  molecular axis of the molecule at xa, then both
  na and -na contribute to the order (i.e.
  quadrupolar not dipolar order) any order
  parameter must be even in na, and a vector order
  parameter is insufficient.
• Try a second rank tensor Q

26
Order Parameter and Order Parameter Tensor
contChaikin and Lubensky

• The order parameter tensor Q can be written in
  terms of the scalar S.
• In a coordinate system with one axis along the
  direction of molecular alignment, the matrix ltQgt
  is diagonal
• ltQijgt S (ninj- 1/3dij)
• Q has the properties that its trace is zero, but
  in the nematic phase ltQgt? 0 (unlike the
  isotropic).

• In general one can expect the degree of order to
  be dependent on temperature.
• Later we will construct models which allow us to
  look at where the phase transition between
  isotropic and anisotropic phases occurs.
• In general it will occur at a specific value for
  S (or equivalently P2.).

Data and theory for p-azoxyanisole PAA
o neutron diffraction data NMR data
27
Fluctuation-Dissipation TheoremDill and
Bromsgrove

• We have seen that the velocity autocorrelation
  function indicates how fast a particle 'forgets'
  its initial velocity due to the effect of
  Brownian motion and collisions leading to
  randomisation.
• If this timescale is long, then clearly there is
  little dissipation, there are few collisions, and
  equilibrium is slow to be achieved.
• Thus low dissipation means it takes a long time
  to establish equilibrium.
• t is m/ ? and is typically in the picosecond
  range for a small protein.

• Conceptually, the fluctuation-dissipation theorem
  states that the fluctuations in a system are
  correlated (inversely) with the energy
  dissipation, so generalises our conclusion from
  the Einstein equation.
• In the Einstein relation
• we see that the drag coefficient ? (dissipation)
  and the diffusion coefficient D (directly related
  to the fluctuations in mean position as we have
  just seen) are inversely related.
• Large dissipation leads to small fluctuations
  about equilibrium.

28
Fluctuation-Dissipation Theorem cont

• Conversely, if the area is large, the dissipation
  is small, and the fluctuations are large.
• So we have a statement about an equilibrium
  property the fluctuations related to a
  non-equilibrum property , in the form of
  dissipation.
• The magnitude of equilibrium fluctuations is
  related to how fast the system reaches
  equilibrium.

• Integrate the time correlation function over all
  possible lag times t
• This integral equals the area under the curve.
• If the area is small it implies that kT/? is
  small and D is also small the dissipation is
  large, and the diffusion coefficient/ transport
  coefficient is small.
• Equilibrium is rapidly reached as there are many
  collisions.
• And the fluctuations about equilibrium are small,
  as we saw from the MSD.

29
Further thoughts on the Fluctuation Dissipation
Theorem cont

• Further Examples of how the FDT applies
• Consider a pendulum moving about an equilibrium
  position, due to fluctuations in air compared
  with its motion in a viscous fluid.
• At a fixed temperature, the mean square
  displacement will be the same in both cases,
  corresponding to the magnitude of the fluctuation
  in the displacement.
• However since the damping in the second case is
  much greater, then the random forces f(t) giving
  rise to the fluctuations must be greater too.
• An alternative form of the equation expressing
  this relationship is given by
• force correlation function

• Nyquist's formula in electrical circuits says
  that the larger the resistance (giving rise to
  dissipation) the larger the (thermal) noise emf
  present.
• Thus this theorem translates into many different
  situations where fluctuations and dissipation are
  present.
• BUT It only applies to equilibrium systems.
• Recent work has been directed at trying to
  understand how far it can be pushed e.g. for
  glasses which have quenched in disorder

30
Fluctuations in Membranes treated as a Simple
FluidSafran

• Hence, writing the undulations as a Fourier sum
• The free energy DF is given by
• And therefore by equipartition of energy (for
  each mode q)

• For a simple fluid surface there will be an
  associated surface energy g per unit area.
• The change in free energy associated with the
  undulations of the surface h(x,y) is
• 
  where

31
Interpreting these FluctuationsSafran

• Thus the mean square fluctuation diverges as the
  logarithm of the system size.
• Also, since
• small wavelength fluctuations in real space (ie
  large q) have far smaller fluctuations associated
  with them than large wavelength (small q)
  fluctuations.
• Or equivalently higher energy is associated with
  small wavelength fluctuations.
• Thus largescale distortions of membranes are more
  favourable than local rumpling.

• Consider
• In two dimensions the logarithm of the integral
  diverges at small q, and so we must think
  carefully about the limits.
• Set the upper limit as p/a, where a is the atomic
  size.
• Set the lower limit as p /L, where L is the size
  of the interface.
• Then

32
Real Cells

• Example Bovine pulmonary arterial endothelial
  cells imaged in the confocal laser scanning
  microscope
• Dual labeled with a green stain for actin
  microfilaments (FITC-Phalloidoin) and a red stain
  for mitochondria.
• The stain for actin is actually derived from a
  toxic compound found in mushrooms.
• The stain for mitochondria only becomes
  fluorescent when the stain is activated by
  enzymes which reside in the mitochondria.
• For this reason, only the mitochondria appear
  red, not the other organelles.
• Example Bone cells imaged in the environmental
  scanning electron microscope (ESEM).
• Normal scanning electron microscopes work in high
  vacuum, and so require cells to be dehydrated.
• The ESEM does not, and so potentially provides a
  new route to high resolution examination of
  cells.