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How does it work? A liquid is subjected to a monochromatic beam (fixed wavelength) that has been collimated so all rays are parallel and in phase 3
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1
--Experimental determinations of radial distribution functions
--Potential of Mean Force
2
Radial distribution function from experiments
• A diffraction experiment uses radiation of a wavelength < the molecular size
• For example X-Ray scattering (l = 0.01 to 10 Å) or neutron beams (l = 1 to 10 Å)
• How does it work? Electrons of an atom or molecule do the scattering in X-Ray (needs an X-Ray generator) ; while in neutron scattering the nucleus of the atom is the scattering center of neutrons (needs a neutron beam source)
3
How does it work?• A liquid is subjected to a
monochromatic beam (fixed wavelength) that has been collimated so all rays are parallel and in phase
4
Reflection scattering experiment
The scattered radiation is measured as a function of the scattered angle
5
Scattering is defined by the vector s
l
l
sin4
)(2 0
s
SSs
In X-Ray, the electrons are the scattering sites, and the scattering cross section isrelated to the Fourier transform of the electron density:
rdersf rise
.)()(
can be calculated and it is known for most atoms
6
the experiment measures the intensity of the scattered radiation at each
scattered angle
difference of the path length of the two scattered rays is given by a distance x2 –x1
21.
21210
21012
)()()(
. ).(2 ).(
risesFsfsA
rsrSS
rSSxx
l
amplitude of scattered radiationat the angle that corresponds to s
7
Intensity of the scattered radiation
drrgesfNsNfsI
sAsIris )()()()(
)()(.22
2
N is the # of scatters in the target region of the liquid that is subjected to the beam(not known); so is normalized to the scattering of atoms without interference
dresfdrrgesfsfsI risris .2.22 )(]1)([)()()('
8
the final diffraction equation isdrr
srsrrgsfsfsI 2
0
22 )sin(]1)([)(2)()('
or total structure function
sdssrsHr
rgrh
drrsrsrrgsH
sfsfsI
02
2
02
2
)sin()(2
11)()(
1- g(r)function n correlatio totalthe of ansformFourier tr theis H(s)
)sin(]1)([2)()(
)()('
9
Intensity of scattered radiation for liquid Ar at T= -125oC and 0.982
g/cc90% confidence interval
10
Structure function H(s) from the X-ray diffraction data
11
Radial distribution function computed from the experimental
H(s)
uncertainty shown by the hatched region
12
Potential of Mean Forceat low densities
forcemean of potential theis wwhere),,(
:densities allat )0,,(
/)(12
/)(12
12
12
kTrw
kTru
eTrg
eTrg
--w is the effective potential between two particles modified by the presence of all other particles;
--range of w (r12) much longer than that of u(r12);
--w12 is a function of density and temperature (unlike u(r12))
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more formal expression of the PMF
12
1212
)(rdrduF
force between two atoms in vacuum as they move apart
in a fluid
2,112
321
12
1212
),...,,()(
rr
N
rdrrrrdu
rdrdwF
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relation of PMF to g(r)
),(ln
........
....),...,,(....
),...,,()(
2112
12
43/
4312
321/
2,112
321
12
1212
rrgrddkTF
rdrdrde
rdrdrdrd
rrrrdue
rdrrrrdu
rdrdwF
NkTu
NNkTu
rr
N
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Relation of PMF to g(r)
),(ln),(0 c so
0 wand 1 g(r) separated, infinitely are 2 and 1 atomswhen ),(ln),(
gIntegratin
),(ln),(
2121
2121
211212
2112
rrgkTrrw
crrgkTrrw
rrgrddkT
rdrrdwF
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physical interpretation of PMF
since w(r12,T,) is the integral of the force over the distance, it is also the work done to bring two particles together from an infiniteseparation in a dense fluid to a separation r.
Since this work is done at N, V, and T then w(r) isthe Helmholtz free energy of the process
17
g(r) and w(r) for the Hard
spheres fluid:--Range of PMF is longer than
range of u
18
--Note that the PMF shows
attractive potential when the
HS is purely
repulsive, why?
19
2 atoms sufficiently close to each other; on each collision with surrounding atoms, the force is indicated by arrows; there is a region shielded from collisions with other atoms. Due to this imbalance there is a net attractive force between the two atoms at these distances
20
Behavior of polymers, colloids, or proteins in solution
• one possibility of obtaining the rdf for these systems (besides molecular simulations) is to develop an expression for he PMF and then fit the parameters to experimental data, for example osmotic P or precipitation data
• Since the solvent molecule is small compared to the macromolecules, solvent is treated as a continuum
21
Example: precipitation of globular proteins in aqueous solution induced by a polymer PMF model:
),,()()(),,( TrwrwrwTrw electrattHS
Attractive term: Van der Waals, for example 12-6 LJ:
6
6
)/(36)(
)(
rHrw
rCrw
att
att
H is called the Hamaker constant
22
Example: precipitation of globular proteins in aqueous solution induced by a polymer
• For the electrostatic term:charge-charge, charge-dipole, charge-induced dipole if the molecules are charged, but modified by the presence of the solvent:
22
)(2
4length Debye theis
)1()(
ii
i
r
elect
qkT
reqrw
23
Hamaker
electrostatic
pmf
for this set of parameters, the potentialis attractive for r/ > 1.5
24
for this set of parameters, the potentialis always repulsive
25
for this set of parameters, the potentialis attractive for r/ > 1.1
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for this set of parameters, the potentialis attractive for r/ > 1.56
27
Example: precipitation of globular proteins in aqueous solution induced by a polymer
• An attractive force arises due to the exclusion fo the polymer from the region between two macromolecules; this is added as an osmotic term:
• the osmotic term depends on the size of the polymer
),,(),,()()(),,( TrwTrwrwrwTrw osmelectrattHS
28
Osmotic pressure and PMF for colloidal and protein solutions
• The Virial EOS was derived for a dilute concentration of atoms in vapor phase, i.e., space between atoms is vacuum.
• Another Virial EOS can be derived considering the solvent as a continuum fluid where the molecules are floating. The solvent is chracterized by T, P, dielectric constant, and chemical potential m.
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Osmotic pressure and PMF for colloidal and protein solutions
• Considering the addition of solute to the solution at constant T and chemical potential of the solvent, m.
• As solute is added, the equilibrium pressure above the solution increases to keep the chemical potential constant (that is changing due to the addition of the solute)
30
Osmotic pressure and PMF for colloidal and protein solutions
At moderate solute concentrations,
...),(),(1 232 solutesolutesolutesolventsolution TBTBkTPP mm
31
Osmotic pressure and PMF for colloidal and protein solutions
• It is important to understand the difference of the B2 in gases vs. B2 for solutions; in the first case B2 depends on u(r) but B2 of solutions depends on w(r, , T)
• The B2 values can be obtained from osmotic pressure measurements. If the values are negative (positive), the net force is attractive (repulsive).
• The sign of the 2nd osmotic Virial corfficient gives hints regarding whether the protein is going to precipitate (crystallize) or not.