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))

13

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

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--Note that the PMF shows

attractive potential when the

HS is purely

repulsive, why?

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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

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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

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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

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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

26

for this set of parameters, the potentialis attractive for r/ > 1.56

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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

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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)

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Osmotic pressure and PMF for colloidal and protein solutions

At moderate solute concentrations,

...),(),(1 232 solutesolutesolutesolventsolution TBTBkTPP mm

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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.