Dynamics of Non Equilibrium Fluctuations in Free Diffusion

 8
procedure led to the writing of a custom-made software able to extract the temporal 
correlation function of the investigated sample out of the images. 
Experiments have been performed on free-diffusing samples and the temporal 
correlation function of non-equilibrium fluctuations have been recovered as a function 
of the wave vector showing the non-diffusive decay behavior below the critical wave 
vector thus confirming for the first time theoretical predictions and giving evidence 
that the temporal correlation function of non-equilibrium fluctuations is an exponential 
decay within experimental accuracy. 
Other experiments are foreseen as a development of this work to investigate the 
dynamics of non-equilibrium fluctuations in different diffusive processes like that of a 
single fluid under the effect of a temperature gradient or a binary mixture in the same 
conditions to analyze the behavior of both thermal and concentration fluctuations. 
Moreover it will be very much interesting to analyze the dynamics when analogous 
experiments will be performed in a micro-gravity environment, like in the 
GRADFLEX experiment that is scheduled to be flown on the Foton M3 Russian 
vector. 
The dynamical technique here introduced is well suited to study the dynamics of slow 
processes, therefore future experiments can be foreseen on gels or glass-like systems. 
The remainder of the present work is organized as follows: in chapter 2 we will 
theoretically calculate the structure factor of non-equilibrium fluctuations in an 
isothermal binary mixture undergoing free diffusion. 
In chapter 3 we will first derive the transfer function of the shadowgraph technique 
and then that of a family of optical techniques ranging from the Schlieren to the 
shadowgraph. 
In chapter 4 we will explain in details the new procedure and describe the algorithms 
we implemented to extract the static and dynamic data from images. 
In chapter 5 we will describe the experimental set-up. 
In chapter 6 we will present experimental results with comments on them. 
At the end, in chapter 7 we will briefly summarize the entire work. 
 
 9
    
2. Theory of Non Equilibrium 
Fluctuations 
 
 
 
Thermodynamics describes a fluid as the statistical ensemble of little volumes 
characterized by physical variables such as velocity, temperature, density and so on. 
Even in the equilibrium state these quantities are not constant in time but fluctuate 
close to an average value. The auto-correlation of these fluctuations is a possible way 
to investigate physics within fluids because it’s related to the intensity of light 
scattered by the fluid. For example fluctuations in the equilibrium state originate the 
scattering of light by the fluid. 
In this chapter we will derive the dynamic structure factor of concentration 
fluctuations in an isothermal binary mixture under the influence of gravity and of a 
time-dependent concentration gradient. We will follow the guidelines of Landau’s 
linearized hydrodynamics [1], as extended by Vailati and Giglio [2] to the case of 
time-dependent diffusion processes. Two main results will be derived. 
First we will derive the static power spectrum of non-equilibrium fluctuations as a 
function of their wave vector q . We will see that the power spectrum diverges with a 
4−q  dependence as the wave vector is decreased. This divergence is prevented for 
wave vectors smaller than a typical value cq  depending on fluid properties and 
gravity. The effect of gravity sets the power spectrum to a constant value for small 
wave vectors. 
The second result is the analysis of the dynamics of non-equilibrium fluctuations, i.e. 
their temporal evolution. The temporal correlation function will be derived to be, 
under appropriate conditions, well described by an exponential decay with 
characteristic time τ  depending on fluctuation wave vector. The time constant appear 
to be equal to ( )21 Dq , i.e. the diffusive one, for wave vectors larger than the cut-off 
value cq , while it is proportional to 2q  for wave vectors below the cut-off value, due 
to the effect of gravity. The effect of gravity can be summarized as follows [3]; a 
fluctuation is but a small volume of the stratified fluid, which moves parallel to the 
concentration gradient, i.e. vertically, originated by a velocity fluctuation in the fluid. 
The displaced volume comes to a layer of fluid with different density and 
concentration, and is subjected to two different phenomena: diffusion trying to relax 
the concentration gradient and buoyancy trying to move the volume back to the iso-
picnic layer. These two phenomena have time constants depending in different ways 
on the fluctuation size. So the fluctuation behaviour is different depending on 
fluctuation size. 
For large wave vectors with respect to the cut-off one, the fluctuation is small, then 
diffusion relaxes the concentration gradient very quickly before the buoyancy force 
given by gravity can play a role. In this regime, which we call diffusive, the power 
 10
spectrum exhibits the 4−q  divergence as the wave vector is decreased, while the 
decay time is the usual diffusive one ( )21 Dq . 
For small enough wave vectors, the fluctuation is big and diffusion becomes too slow, 
therefore gravity moves the fluctuation back to a layer with same density. The effect of 
this is that the power spectrum becomes independent of wave vector and the 
correlation time becomes proportional to 2q . So gravity makes big fluctuations 
disappear more quickly than they would have done without its effect and moreover it 
makes the concentration contribution of a big fluctuation much smaller. We call this 
regime gravitational. 
 
 
2.1. Fluctuating Hydrodynamics 
 
Fluctuations in simple fluids in a homogeneous equilibrium state have been studied 
since the first decades of 20th century by Einstein [4] and their results have been 
derived in alternative and simpler way by Landau and Lifshitz at the end of the 50’s 
through fluctuating hydrodynamics [1], a linear theory including stochastic terms 
describing the noisy spontaneous generation of fluctuations. 
In typical light scattering experiments one has direct access to the dynamic structure 
factor ( )ω,qS , which is the power spectrum of refractive index fluctuations at wave 
vector q  and frequency ω . Fluctuations power spectrum for a simple fluid in a 
homogeneous equilibrium state is the product of three distinct Lorentzian curves: one 
is centered at 0=ω , which is called Rayleigh line and whose intensity is independent 
of wave vector q  and is generated by a thermal diffusive mode for spontaneous 
temperature fluctuations; the other two lines are symmetrically shifted in frequency 
and are called Brillouin lines, being caused by propagating modes within the fluid. 
The first description of non-equilibrium fluctuations is due to Kirkpatrick et al. [5], 
who applied the kinetic theory with mode-mode coupling to analyze the effect of a 
temperature gradient in a fluid and showed that the Rayleigh line becomes the 
superposition of two Lorentzian curves originated by two visco-heat modes due to the 
coupling of temperature fluctuations and velocity ones, driven by the presence of the 
macroscopic gradient.  
Also the overall intensity of the Rayleigh line was greatly increased and was derived 
to be proportional to 42 qT∇  showing that the coupling of velocity fluctuations to 
the imposed gradient can give rise to long-ranged spatial correlations within the fluid. 
In the following years the simpler fluctuating hydrodynamics approach had been 
applied by Ronis and Procaccia to the study of fluids and fluid mixtures outside 
thermo-dynamical equilibrium with equivalent results [6]. 
Further work was done in the 90’s to include the effect of gravity in non-equilibrium 
steady state systems and showed that the buoyancy force saturates the 4−q  divergence 
for very small wave vectors. These results have been derived both for simple fluids [7] 
and for binary mixtures under the combined effects of a steady state temperature 
gradient and the Soret-effect-induced concentration gradient [8]. 
 11
Only at the end of the last century these results have been extended to the case of time-
dependent diffusion processes [2], which can well describe the transient state when a 
Soret concentration profile is built up or a free diffusion process, where an initial 
inhomogeneous concentration profile relaxes to the homogeneous state. 
Another important issue has been recently addressed by the group of de Zarate [9] and 
is that of the boundary conditions, which for extremely small wave vectors cannot be 
neglected giving rise to a net decrease in the intensity of fluctuations larger than the 
sample cell thickness, but for all the experiments reported in the present work this 
effect is negligible. 
In the present work we are interested in free diffusion processes where the temperature 
is approximately homogeneous allover the fluid so we will consider only concentration 
fluctuations and omit terms due to temperature fluctuations which give negligible 
contributions to the scattering intensity. 
We will consider the case of an isothermal binary mixture of two miscible fluids 
subjected to the gravity force g


 with the denser fluid at bottom (stable configuration) 
and a gradient of the concentration of the denser fluid parallel to the gravitation 
acceleration g


. The concentration gradient ( )tzc ,∇


 is supposed to depend on the 
height z  and time t . The linearized hydrodynamic equations governing the fluid can 
be written as [2,8]: 
 











+∇+∇−=
∂
∂
⋅∇−=∇⋅+
∂
∂
gup
t
u
jcu
t
c











21
1
ν
ρ
ρ
 Eq.2.1 
 
in which c  is the mass fraction of the denser component, u
  the fluid velocity, ρ  the 
mass density, j


 the mass diffusion flux, p  the pressure and ν  the kinematic 
viscosity.  
The first equation expresses mass conservation, while the second one expresses 
momentum conservation. 
To linearize the equations we write each hydrodynamic variable as the sum of its 
average value plus a fluctuating term, which is supposed to be small with respect to 
the average value: 
 
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )







=+=
+=+=
+=
txutxutxutxu
txctxtxtxtx
txctxctxc
,,,,
,1,,,,
,,,



















δδ
βδρδρρρ
δ
  Eq.2.2 
 
in which ( ) Tpc ,
1
∂∂= − ρρβ  is the solutal expansion coefficient and x


 is the two-
dimensional position vector lying in a plane orthogonal to the concentration gradient. 
We have assumed that the only relevant density fluctuations are given by 
 12
concentration ones and that the fluid is supposed to be (on the average) at rest 
( ) 0, =txu



. 
Substituting Eq.2.2 into the hydrodynamic equations Eq.2.1 and omitting the second 
order terms, we get: 
 










⋅∇+∇+∇+
∂
∂
−=−∇
⋅∇+⋅∇+⋅∇−∇⋅−
∂
∂
−=⋅∇+
∂
∂
Supc
t
u
gp
Fjcjcu
t
c
j
t
c




















ρ
δνβδ
ρ
δ
ρ
βδ
ρ
δ
ρ
δ
δ
ρ
111
111
2
 Eq.2.3 
 
in which we have added the random forces F


 and S  accounting for the spontaneous 
onset of the concentration and velocity fluctuations respectively. 
Two additional equations are needed to describe the system. First the diffusion 
equation giving the time evolution of the macroscopic concentration profile: 
 
0
1
=⋅∇+
∂
∂
j
t
c 


ρ
 Eq.2.4 
 
Second the equation describing the hydrostatic pressure gradient due to the presence of 
the gravitational field: 
 
0
1
=∇− pg



ρ
 Eq.2.5 
 
Inserting Eqs.2.4-2.5 into Eq.2.3 leads to the system: 
 











⋅∇+∇++=
∂
∂
⋅∇+⋅∇+⋅∇−∇⋅−=
∂
∂
Sucg
t
u
Fjcjcu
t
c















ρ
δνδβ
δ
βδ
ρ
δ
ρ
δ
δ
1
11
2
 Eq.2.6 
 
in which the term cu ∇⋅



δ  in the first equation represents a source term for the 
concentration fluctuations originating from a velocity fluctuation, while the term 
cgδβ


 represents the corresponding source of velocity fluctuations given by a 
concentration fluctuation, due to the presence of the gravity force. 
The mass diffusion flux is in general described by the phenomenological relation: 
 






	


∇+∇+∇−= p
p
k
T
T
k
cDj pT





ρ  
 
 13
in which D  is the diffusion coefficient, Tk  is the thermal diffusion ratio, T  is the 
absolute temperature pk  is the baro-diffusion ratio and p  is the pressure. The first 
term is given by the Fick flow due to diffusion, the second one is induced by the Soret 
effect while the third is given by the sedimentation flow. The latest term, given by 
sedimentation, is negligible with respect to the other two (actually it’s important only 
if the other two terms do vanish). In the case of free diffusion only the first term 
related to the Fick flow is present: 
 
cDj ∇−=



ρ   Eq.2.7 
 
The fluctuating part of the mass flux can be written as a function of thermodynamic 
variables in the form: 
 
ccDjcjccDj ∇⋅∇−⋅∇+⋅∇+∇−=⋅∇









δρβδββδδρδ 2  Eq.2.8 
 
After including Eq.2.8 into the system Eq.2.6, the hydrodynamic equations become: 
 










⋅∇+=∇−
∂
∂
⋅∇+∇=∇⋅+
∂
∂
Sgcu
t
u
FcDcu
t
c











ρ
βδδν
δ
δδ
δ
12
2
 Eq.2.9 
 
To get the power spectrum of the concentration fluctuations we perform a space-time 
Fourier transform of Eq.2.9 as defined by: 
 
( )[ ]{ }txqitzcxddttzcq ωδδ ω −⋅= 





 exp),(),(
,
 
 
in which q
  is the bi-dimensional spatial wave vector, while ω  is the temporal 
frequency. We get: 
 
( )
( )






⋅⋅−⋅=+
⋅−∇⋅−=+
PSq
i
Pgtzcqitz
FqitzctzDqitzc
qqq
qqq
ωωω
ωωω
ρ
βδνωδ
δωδ
,
,
2
,
,,
2
,
),(),(v
),(),(v),(

















 Eq.2.10 
 
in which ),(v
,
tzq ωδ 



 is the transverse velocity fluctuation as defined by: 
  
Ptzutz qq ⋅= ),(),(v ,, ωω δδ 





, 
 
being P  the projector in the direction perpendicular to the concentration gradient: 
 
qqP ˆˆ1−=  
 14
 
and being qqq


=ˆ  the unitary vector with direction q
 . 
Solving the system Eq.2.10 we finally get: 
 
( )
( ) ( )[ ]cgqiDqi
cPSq
i
Fqqii
c
qq
q
∇⋅+++
∇⋅⋅⋅−⋅+−
=













βνωω
ρ
νω
δ
ωω
ω
22
,
,
2
,
  Eq.2.11 
 
in which we drop the dependence on height z  and time t . 
To get the correlation of concentration fluctuations we must assign the correlations of 
the noise terms F


and S , which are assumed to retain the same correlations as in the 
equilibrium state [2]: 
 
( ) ( )
[ ] [ ]



















=
∇=∇⋅⋅⋅∇⋅⋅⋅
′−′−




	


∂
∂
=
′′
′′
0
8
8
,
*
,
2
4
,
*
,
,
,
4
,
*
,
ωω
ωω
ωω
ρν
pi
ωωδδδ
µρpi
qq
B
qq
ji
Tp
B
q
j
q
i
SF
c
Tk
cPSqcPSq
qq
cDTkFF

















  Eq.2.12 
 
At the end the correlation properties of concentration fluctuations are: 
 
( ) ( )
( )( )












∇⋅+++
∇+∂∂+
=
2
22
22
,
2422
4
,
*
,
8 cgqiDqi
cqcDqqTk
cc
TpB
qq 





βνωω
νµνω
ρpi
δδ ωω   Eq.2.13 
 
The denominator of Eq.2.13 can be factorized as ( )( )2222 −+ Γ+Γ+ ωω , where the 
two roots 2+Γ  and 2−Γ  can be written as: 
 




















	








	


+



	


+−



	


+±






	








	


+−



	


+=Γ±
4244224
2 1141121
2 q
qDDD
q
qDDq cc
ννννν
ν
 
 Eq.2.14 
 
in which we have defined the cut-off wave vector: 
 
( )
( )
4
,
,
D
tzcg
tzqc
ν
β ∇
=   Eq.2.15 
 15
 
The physical meaning of the cut-off wave vector will be clearer when we’ll analyze 
the dynamics of fluctuations, but essentially it represents the wave vector at which the 
effect of gravity starts being important or, alternatively, the wave vector separating the 
diffusive regime from the gravitational one. The cut-off wave vector as defined by 
Eq.2.15 depends on fluid properties as well as on the strength of the gravitational field. 
Therefore it can be significantly decreased if the gravity level is decreased for example 
in a orbital experiment, in which gravity can be reduced by a factor of 610− . 
The power spectrum becomes: 
 
( ) ( )
( )( ) ( )( )











Γ+Γ+
∇
+
Γ+Γ+
∂∂+
=
−+−+
2222
22
2222
,
2422
4
,
*
,
8
ωω
ν
ωω
µνω
ρpi
δδ ωω
cqcDqqTk
cc
TpB
qq



 
 Eq.2.16 
 
The spectrum is therefore the sum of two terms each being the product of two 
Lorentzian curves with decay rates +Γ  and −Γ , which can be re-written in the form: 
 




















	








	


+−



	


+±



	


+=Γ±
422
1411
2 q
qDDDq c
ννν
ν
  Eq.2.17 
 
The argument of the squared root is positive if we met the condition: 
 
( )
ccp q
D
q
D
D
qq 44
2
44
νν
ν
≅
−
=>   Eq.2.18 
 
in which we have defined the quantity pq  as the wave vector at which the propagating 
modes are present in the system. For wave vectors smaller than pq , in fact, the 
argument of the square root becomes negative and the decay rates +Γ  and −Γ  acquire 
an imaginary part corresponding to the propagation of a wave within the fluid. In this 
regime for pqq <  fluctuation are over-stabilized by the buoyancy force. 
It’s very useful to analyze in detail the form of the two decay rates whose real and 
imaginary part are displayed in graphs in Figs.2.1-2.3 after division by 2q . 
 16
  
 
Fig.2.1 Real part (left figure) and imaginary part of the decay rates +Γ  (black line, viscous one) and −Γ  
(red line, mass diffusive one) divided by 2q  for a 40% w/w glycerol in water vs. pure water binary mixture 
undergoing free diffusion. Here scm0166.0 2=ν , scm1093.0 25−= xD  and 1cm150 −=cq , 
therefore from Eq.2.18 we get 1cm33 −≅pq . 
 
  
 
 Fig.2.2 Same as Fig.2.1, but for wave vectors close to pq . 
 
 
 
Fig.2.3 Same as Fig.2.1 (real part only), but in a Log-Log plot.