Einstein and 1905
(2005 was Einstein Year – celebrating 5 key papers published
that year.)
•
In 1905 the atomic hypothesis was
not fully accepted.
•
Despite Brownian motion having
been known about for 75 years, its
significance was not appreciated.
•
The kinetic theory of gases was
thought of as a 'mechanical
analogue', but implied reversibility.
•
The 2nd law of thermodynamics
required irreversibility.
•
Einstein understood that taking a
statistical approach, and assuming
atoms existed, reconciled the
paradox.
•
His paper, the second of the 5, was
written in April 1905.
•
•
•
A macroscopic particle – such as the
pollen particle – would be buffeted by the
atoms/molecules in the surrounding water.
The particle would undergo diffusion and
measuring the diffusion constant (or
equivalently the displacement) should
show an increase with t (not linearly).
Perrin's subsequent experiments on
sedimentation showed how all this hung
together.
34
Brownian Motion
Nelson; Dill and Bromberg
•
•
•
•
Brownian motion of a particle arises due
to its constant bombardment by
molecules(e.g as in the first observation
by Brown in 1827 of pollen grains by
the water molecules).
Net force, averaged over time, on the
particle is zero.
But at any moment there is a constantly
fluctuating net force and this gives rise
to the observed Brownian motion.
The motion of the particle follows a
random walk
•
•
•
•
For a sphere of radius r in a liquid of
viscosity , we have seen from Stokes
Law the force is 6prv
This can also be written in terms of the
drag coefficient, defined so that
Force = drag coefficient × velocity
x= 6pr
Equation of motion for particle of mass m
d 2R
dR
m 2 x
 Frandom
dt
dt
where R(t) is the position coordinate, and
Frandom is the random force due to
collisions.
35
Brownian Motion
Jones
•
Forces are random so each
direction behaves in the same way
and
•
We can also use the identity
2
d  dx   dx 
d 2x
x     x 2
dt  dt   dt 
dt
 x 2  y 2  z 2 
 R  3  x 
2
•
2
Work in 1D for simplicity
d 2x
dx
m 2 x
 Frandom
dt
dt
•
Use the substitution
d (x2 )
dx
 2x
dt
dt
•
Then
d 2 x x d (x2 )
m 2 
 Frandom
2 x dt
dt
•
Thus
2
d  dx 
 dx  x d ( x )
m  x   m  
 xFrandom
2
dt  dt 
dt
 dt 
2
•
Rearranging and taking an average
yields
x d  (x2 ) 
2
dt

d
dx
 dx 
xF
 m
x
  
random
 dt
dt
 dt 

2




36
Brownian Motion cont
x d  (x2 ) 
2
dt

•
d
dx
 dx 
xF
 m
x
  
random
 dt
dt
 dt 

•
•
•
2




The direction of the random force on
the particle is not correlated with the
particle's position, so the first term is
zero. Similarly, there is no
correlation between the particle's
position and its velocity, so the
second term is also zero.
The third term can be rewritten via
the theorem for equipartition of
energy, since
mvx2/2=kBT/2 or vx2=kT/m

d x 2 
dt

And hence the total squared
displacement in 3D is
R 
2
•
3 x
2

6k BT
x
t
The motion is diffusive with a
diffusion coefficient given by the
Einstein relation
D=kBT/x
•
We will explore the implications of
this later, but note the inverse
relation between drag (x) and the
diffusion constant D.
2k BT
x
37
Aside on Ensembles
(for the theoretically inclined)
Waldram:The theory of thermodynamics
•
•
•
•
•
•
On the previous slide we saw averages
<….>.
What are we averaging over?
Those of you who took TP1 will
probably assume this is a time average
– and it may be.
But the ensemble average can also be
over a set of replicas at a given instant.
In your Thermal Physics course you
have been taught about the different
types of ensembles.
You have also been taught about the
Principle of Equal Equilibrium
Probability,namely that for an isolated
system, all microstates compatible with
the given constraints are equally likely
to occur.
•
•
•
•
•
If the system obeys this Principle, then an
average over the probabilities of the
system being in a particular configuration
is the same as the average over time.
But, formally, one should always specify
what sort of average is being taken by the
<…..>.
Frequently is is simply referred to as an
ensemble average, which could mean
either.
Underlying all this is the ergodic
assumption – that the system can go
anywhere within the allowed energy range.
In which case in equilibrium, the
fluctuation distribution is identical with the
ensemble average distribution.
38
The Diffusion Coefficient
•
•
When the diffusing particle is a sphere,
Stokes Law gives
x=6pr, so
k BT
D
6pr
•
•
•
•
This is known as the Stokes- Einstein
equation.
For other shapes the drag coefficient,
and hence diffusion constant, take
different forms.
E.g. disc moving randomly
x = 12r
Ellipsoid (major and minor axes a and b)
x=6pa/ ln(2a/b) for random rotation
(other expressions for motion sideways
or lengthwise)
•
•
•
•
For a molecule such as a protein, the
radius R may not equate to the actual,
physical size.
A hydrodynamic radius RH can be
defined, which is the effective radius
presented to the fluid by the molecule.
Additionally, the molecule may or may
not permit the fluid to drain through it,
depending on the density of chain
packing.
Thus the detailed hydrodynamics of e.g.
globular proteins diffusing through a fluid
can be complex.
We will return to these concepts when
discussing proteins and polymers in more
detail.
39
Typical values for D in water
Molecule
T(oC)
MW
(g mol-1)
D (cm2s-1)
Oxygen
25
32
2.1 x 10-5
Sucrose
25
342
5.23 x 10-6
Myosin
20
493,000
1.16 x 10-7
DNA
20
6,000,000
1.3 x 10-8
Tobacco mosaic
virus
20
50,000,000
3.0x 10-8
Quantitatively one would expect D ~ m-1/3, but because a hard sphere
model for the molecules is not accurate, this precise dependence is not
found.
40
Diffusion Equation
Dill and Bromberg
•
Fick's first law states that the flux J of
particles is proportional to the
concentration gradient.
J  Dc
•
•
•
Specific solutions depend on
boundary conditions (recall your
2nd year maths!).
e.g for point source diffusing in 1D
along x, starting at x=0
And conservation of particles requires
c
 .J
t
•
Hence in 1D and assuming D is a
constant, then
  2c 
c
 D 2 
t
 x 
•
which is Fick's second law in its
simplest form.
Numbers
correspond
to values of
Dt
41
Diffusion Equation cont
•
Compare with data from Wall Street
•
•
Distribution of monthly returns for
a 100-security portfolio 1945-1970
•
•
•
Note that D need not be a constant,
although in many simple situations
it is.
If it is not, then the equation must
be modified to allow for the fact
that D can be a function of
position.
c   
 c  
   D( x)  
t  dx 
 x  
Over time stock prices also exhibit a random
walk with drift!
Individual whims can lead to the statistical
movement.
Baseline drift comes from the fact that
overall shares do make a profit….
42
Diffusion Control
• Diffusion may limit:
1.
– Growth of 2nd phase particles
– Supply of nutrients to organisms
– Colloidal aggregation
a
•
•
•
We will model as spherically symmetric,
so work in spherical coordinates.
Assume steady state so dc/dt=0.
Diffusion equation becomes
1 d 2 (rc)
 c
0
2
r dr
2
•
Solution depends on boundary
conditions.
Diffusion of molecules which react at
the surface.
If these are transformed/lost during the
reaction, then c(r = a) = 0. If the
concentration well away from particle is
c solution is
a
c(r )  c (1  )
r
•
For such a concentration profile, the flux
is given by
dc
Dc a
J (r )   D
 2
dr
r
43
Diffusion Controlled Collisions and Reactions
•
Hence the number of collisions at the
surface per unit time I(a) is
2
•
I (a)  J (a)4pa 2  4pDc a
•
•
•
•
•
Minus sign implies flux is towards
particle (-r direction).
This is the fastest possible rate at which
any process can occur, i.e. when it is
limited by diffusion.
If the reaction at the surface itself is rate
limiting, a process will be slower.
If the diffusing molecule is a nutrient to
an organism, then we also have to think
about its consumption and this will be
proportional to the volume.
Thus if the bacterium etc is too large,
cannot get sufficient supply of the
nutrient by diffusion, and so this will set
an upper limit on size.
•
Growth of particles post-nucleation
In this case the molecules diffusing to
the surface are causing growth of the
particle – a process known as Ostwald
ripening or coarsening.
Flux J(r) as before but now we have
J (r )  
•
1
dr


r2
dt
Integration yields
r3  t
or
r  t 1/ 3
•
This is known as the Lifshitz-Slyozov
law, and is empirically found to hold as
long as diffusion is the limiting factor.
44
Diffusion Controlled Aggregation
•
No salt
•
•
With salt
•
•
•
•
During colloidal aggregation, can have either
diffusion or 'reaction' being the limiting factor.
Which is dominant depends on the conditions
of aggregation
E.g. in the examples shown here of 250nm
polymethyl methylacrylate particles, it is the
salt concentration which is changing.
This changes the interparticle potential.
The shape of the aggregates can provide
insight into the processes occurring.
Very often fractal structures form.
In these the structure is the same at all
lengthscales
R~M
1/ d f
Watching paint
dry; a new paint
formulation; • Where R is the size, M is the mass and the
fractal structures fractal dimension 1<df3 (equality for compact
again visible, but object)
clearly with
45
rearrangement
Aside on Fractals
Jones
•
•
•
• Computer simulations show such fractals should
have a fractal dimension of 1.71 when diffusion
limited aggregation occurs.
• If clusters are allowed themselves to diffuse, the
dimension is larger at 1.78 implying a slightly more
dense structure.
• In practice, the rate at which rearrangements and
sticking occur will all affect the fractal dimension.
• But it always lies between 1 and 3.
•
When aggregation occurs – of
individual molecules or particles –
the structures that result depend on
the probability of sticking
i.e. whether diffusion or sticking is
the limiting factor.
If rearrangements can then occur
the structure will be able to
compact.
If not, a fractal structure results,
over a range of lengthscales from
the size of the particle up to the
size of the aggregate.
46
Implications of the Einstein Relation
Dill and Bromsgrove
•
We have derived the Einstein equation
D
•
•
•
k BT
In this, the drag coefficient x
(dissipation) and the diffusion
coefficient D (fluctuations about
equilibrium) are inversely related.
We will see shortly that this is a very
general result.
In general, if there are random
fluctuating forces f(t) and a drag force
xv acting together on a particle, we can
write
The ensemble average of v is given by
m
x
dv
m  f (t )  xv
dt
•
•
•
•
dv
 f (t )  x v(t )
dt
<f(t)> =0 since this is a fluctuating force
arising from the many collisions.
So the solution for <v(t)> is
v(t )  v(0) exp  x t / m
•
So we can define a characteristic time
t = m/x a viscous relaxation time.
•
Example: For a sphere of radius 10nm
(typical size for a large polymer) in water,
this leads to t ~10-10 s.
This is known as the Langevin equation
(strictly the A-Langevin equation, as
dealing with acceleration).
47
Velocity Correlation Function
(Doi, Balescu – see the latter for a rigorous and general discussion)
•
•
•
•
•
•
More generally, look at the velocity
correlation function defined as
<v(0)v(t)>.
Recall the <…> imply a suitable ensemble
average.
Correlation functions in general measure
the extent to which there is any correlation
between a function at two separate times.
It can be applied to position, force,
velocity etc, and can be defined in the
same way as for the velocity correlation
function.
The larger the value of the function is, the
greater the extent of correlation.
For random motion, the velocity
correlation function equals <v2(0)> for t~0,
and = 0 at large t, when all correlations
lost.
•
•
For so-called stationary stochastic processes,
any autocorrelation function only depends on
the time interval involved
i.e <v(t1)v(t2)> = Cv(t1- t2) with Cv a function
Thus more specifically we can write
v(t )v(0)  2D (t )
•
•
•
Where D is the diffusion coefficient.
This can be seen by considering, the mean
square displacement (MSD) in 1 dimension
about an average position x = 0 for diffusive
motion is
t
t
0
0
t
t
0
0
x(t ) 2   dt1  dt2 v(t1 )v(t2 )
x(t )   dt1  dt2 2 D (t1  t2 )  2 Dt
2
48
Velocity Correlation Function cont
•
Returning to the Langevin equation
m
•
dv
 f (t )  xv
dt
By equipartition of energy
1
1
m v 2 ( 0)  k B T
2
2
kT
 v(0)v(t )  B exp  (x t / m)
m
multiply by v(0) and take the ensemble average
Velocity correlation function
•
dv(t )
 v(0) f (t )  x v(0)v(t )
dt
Now v(0) and f(t) are uncorrelated so that the
first term on RHS = 0, and
d
x
v(0)v(t ) 
v(0)v(t )  0
dt
m
 v(0)v(t )  v 2 (0) exp  (x t / m)
•
<v2(0)>
<v(0)v(t)>
m v(0)
So we have obtained an equation for the
velocity autocorrelation function.
•
•
<v2(0)>e-xt/m
t
The figure confirms that the longer the
time interval, the less the correlation
between v(0) and v(t) .
For timescales shorter than t the
characteristic time) the velocity retains
correlation, but not for time scales much
longer than this.
49
Fluctuation-Dissipation Theorem
Dill and Bromsgrove
•
•
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
D
•
•
•
k BT
x
we see that the drag coefficient x
(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.
•
•
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/ x and is typically in the
picosecond range for a small protein.
50
Fluctuation-Dissipation Theorem cont


0
•
•
•
•
Integrate the time correlation function over
all possible lag times t
<v2(0)>
k BT 
k BT
v(0)v(t ) dt 
exp

(
x
t
/
m
)
dt

D

m 0
x
This integral equals the area under the
curve.
If the area is small it implies that kT/x 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.
<v(0)v(t)>
•
t
•
•
•
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.
51
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
1
x
2k BT


•
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
f (0) f (t ) dt

force correlation function
52
Using Brownian Motion to obtain Local
Viscosities
•
•
•
•
Beads moving around in viscous
fluids provide a means of probing
local viscosities.
In some materials (e.g. liquid
crystals, see later) these may be
anisotropic.
Other systems (e.g gels) may be
heterogeneous.
A generalised version of the Stokes
Einstein equation allows the
complex moduli to be determined
by tracking the motion of the
particles.
53
Using Particle Tracking in Practice
•
mm
The mean square displacement as a function
of time can be followed eg. during some
process.
dk BT
r (t ) 
t
3pa
2
•
mm
mm
•
•
mm
(d is dimensionality, a is particle radius, t is
lag time)
Data for hectorite clay 1 hour and 24 hours
after the start of the experiment, showing how
Brownian motion becomes restricted as the
systems 'sets' or 'gels'.
From the displacements the viscoelastic
moduli can be determined (via Laplace
transforms).
dk BT
G (s) 
3pas r ( s ) 2
54
Chemical Potential and Free Energy
•
Recall the definition of the Helmholtz free
energy F for a simple system
•
F=U-TS
•
•
If one has a reservoir containing N
atoms/molecules, the chemical potential m
is defined as
 F 
m 


N

T ,V
Model situation:
A
•
•
B
Particle (species i) and energy
exchange is possible between two
boxes A and B.
Total change in entropy
The chemical potential is the change in
free energy when one particle moves from
one position, phase etc to another.
Equivalently
 S 
 S 
dS tot   A dE A   B dE B 
 E A 
 E B 
 G 
 U 
 H 
 S 
m 
 
 
  T 


N

N

N

N

T , P 
 S ,V 
S ,P

 E ,V
 S A 
 S B 


i  N dN Ai  i  N dN Bi
 Ai 
 Bi 
•
where summation is over species i.
•
In this field of soft condensed matter, F
and G are often used interchangeably since
the PV term is usually negligible.
55
Chemical Potential and Equilibrium
•
•
Now both energy and particle number
must be conserved, and the net change
in entropy must be zero or greater.
Using the relationship between m and S
and recall 1/T = dS/dE and dEA+dEB=0
this equation reduces to
1 1
 m
m 
  dE A     Ai  Bi dN Ai  0
TA TB 
i 
 TA TB 
•
and this must be true for all the species.
This implies (obviously) that at
equilibrium TA=TB and
•
•
When systems can exchange particles,
not only the temperature but also the
chemical potential for each species
must be equal at equilibrium.
Can now extend all the thermodynamic
functions to include the possibility of
Ni varying.
dF  SdT  pdV   mi dN i
i
dH  TdS  Vdp   mi dN i
i
dG   SdT  VdP   mi dN i
i
m A= m B
56
Aside on the Clausius Clapeyron Equation
•
Liquid
P
Since we must have mgas=mliquid on
the phase boundary, then since dG
= 0 also across the boundary
Solid
Vapour
T
•
Recall this tells us about how
properties change along the phase
boundary, eg between gas and
liquid.
 S gasdT  Vgasdp   SliqdT  Vliqdp
•
But across the boundary T and p
must be equal so
dp S
L


dT V TV
57
Chemical Potential of a Perfect Gas
•
•
Recall F= -kBTlnZ
where Z is the partition function.
•
z zk BT

N
pV
For N indistinguishable particles, as in a
perfect gas,
Z = zN/N!
•
Thus m = mo + kBTlnP
where mo represents the chemical
potential of a standard state.
•
If you have a solution in equilibrium
with its vapour msoln= mvap
•
Hence
•
o
Equivalently m
so ln  m  k BT ln c
•
Where c is the concentration in the
solution.
This result is generally true.
where z is the individual particle partition
function.
•
Now z/N can be written in terms of the
pressure as
Then


 z
F  k BT  N ln    N 
N


Hence
 F 
 z
  k BT ln  
 N T ,V
N
m 
•
m so ln  m o  k BT ln P
58
Membranes
•
•
•
•
•
Membranes are the envelopes
around all our cells in our bodies.
They are complex, dynamic
structures comprising so-called
lipid bilayers (see later).
They have to be able to respond to
significant shape changes as our
bodies move or an organism
crawls.
Initially let us treat the membrane
as a simple fluid interface with
associated surface tension, and look
at its fluctuations.
This is far too simple, because in
fact there is a curvature elasticity to
be taken into account (to be
discussed later).
Vesicles – simple
approximation to
biological cells - easily
distort to allow for shape
changes, but do not
exhibit short wavelength
fluctuations as these cost a
lot of energy.
59
Monge Representation of a Membrane Surface
Boal
•
•
•
The Monge representation is a common
way of representing an undulating
surface, such as a membrane.
A point r on a surface is represented by
r = (x,y, h(x,y))
•
We can construct tangent vectors
along x and y, represented by rx and
ry by taking a unit step in the x (or
y) direction, and a step
h
 hx
x
 h

 or
 hy 
 y

in the z direction, to give
rx  (1, 0, hx ) ry  (0, 1, hy )
z y
x
•
•
h(x,y)
h represents the 'height' away from the
x,y surface.
For complex surfaces, h may not be
single valued (e.g if overhangs), but in
the Monge representation it is
constrained to be single-valued.
•
•
These are not unit vectors, and are
not generally orthogonal.
They define the plane tangent to
the surface at (x,y,h(x,y)) by


r r 
 h ,h ,1
n

x
y
rx  ry
1  h

x
y
2
x
2 1/ 2
y
h
60
Monge Representation cont
•
n rydy
z
hx
x
•
rxdx
1
•
  2r 
C  n. 2 
 s 
The element of area dA is defined
by

dA  rx  ry dxdy  1  h  h
•
dA
2
x

2 1/ 2
y
And the metric g of the surface is
defined by
g  (1  hx2  hy2 )
The surface can also be defined by
its curvature – these are simply
geometrical relationships,
discussed in maths texts.
The curvature of a surface is
defined as
dxdy
•
•
•
where s is the arc length.
In general there will be two
principal curvatures (the two
extremal values) C1 and C2.
The combination (C1+ C2) is the
mean curvature .
And C1. C2 is the Gaussian
curvature .
So that dA=g dx dy
61
Fluctuations in Membranes treated as a
Simple Fluid
Safran
•
h(r )   h(q)e iq.r
h(x,y)
q
h(q)   h(r )e iq.r dr
fluctuating surface
•
•
•
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
g
where
2
2
F 
 (h
x
 hy )dxdy
2
h
h
hx  ; hy 
x
y
Hence, writing the undulations as a
Fourier sum
•
The free energy F is given by
F 
g
q

2
2
| h(q) |2
q
•
And therefore by equipartition of
energy (for each mode q)
k BT
| h(q) |  2
gq
2
62
Interpreting these Fluctuations
Safran
•
Consider
h 2 (r ) 
•
•
•
•
•
1
(2p ) 2

| h(q) |2 dq
•
Thus the mean square fluctuation
diverges as the logarithm of the
system size.
Also, since
k BT
| h(q) |  2
gq
2
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
k T
L
h (r )  B log
2pg
a
2
•
•
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.
63
Real Cells
•
•
•
•
•
Bone 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.
64
Real Cells cont
•
•
Real cells are a lot more complex than simply an interface undergoing
fluctuations, but the basic ideas are correct.
Have to take into account
– The physical structure of the membrane
– The (bio)chemical structure of the membrane
– The contents of the cell, particularly the cytoskeleton.
– What is surrounding the cell
– Osmotic effects
Example: what happens
when the cell is placed in
different media which
cause different degrees of
stress? The example shown is for
changing the osmolarity of the
surrounding medium, when the cell
shrinks or stretches according to
whether molecules have flowed in or
out.
65
Aside on Osmotic Pressure
•
•
•
The comment on osmolarity
introduces the idea of osmotic
pressure.
You will be familiar with this
through the idea of what happens
with a semi-permeable membrane.
A
•
B
•
pV  RT  ci
•
Which for low concentrations approximates to
p
•
•
•
If there is solute in A, then water will
flow across the semi-permeable
membrane until the chemical
•
potentials on the two sides are equal.
m mixture (T , p  p , ci )  m pure (T , p,0)
The osmotic pressure, p, is the pressure which
will just stop the flow occurring.
In general
N
RT
V
This has a form equivalent to the case of a
perfect gas.
This effect is crucial in maintaining cells healthy.
For instance, when a plant 'wilts' it has lost water
from its cells and the external pressure causes the
cells to collapse.
By watering the plant, water is sucked back into
the cells to return the plant to full 'turgor' (in this
context one talks about 'turgor pressure', and the
cells are then said to be turgid).
66
Brownian Motion within Cells- current (and
preliminary!) Research
Two dimensional mean square
displacements of beads in this cell.
A selection of particles have been
highlighted in red, for clarity. The
slopes exhibit significant scatter,
Fluorescence image of an individual
which we attribute to local
human dermal fibroblast cell showing
variations in the rheology according67
clustering of the beads in the perinuclear
region, with a few beads in the periphery of to cytoskeletal arrangements.
(Preliminary) Microrheology
It is clear very different responses can
be measured.
From the mean square displacements
one can then extract the local moduli,
and examine how anisotropic they
are.
The aim is now to relate these local
measurements to the internal
components of the cell under different
conditions e.g of cell adhesion,
motility, stress etc.
We do not yet know if the variation is
real, and we will be able to make
these correlations, or simply due to a
spread in values for statistical
68
reasons.