高分子物理 课件 Polymer Physics I

Expectations, Outcomes and Concepts
For Polymers R ~ N
~ N Elasticity/entropy relationship
10 Polymer chemistry
Polymer chemistry is predominantly a
branch of organic chemistry: most
polymers are based on carbon.
The simplest chemical structure is a
carbon main chain with two hydrogen
atoms per carbon-polyethylene.
Consider only linear chains
N>>1, very large
atactic arrangement involves quenched
disorder
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(i). the energy difference between the minima is (ii). The energy barrier is E When is less than k B T, the chain is
statically flexible.
The relative weight of the conformations is of order unity so the chain is not a rigid rod.
<k B T defines the limit of extreme flexibility
k B 1 the trans state is preferred: locally the chain will be rigid For polyethylene, l
function of where
exp(k B for >0
~ Angstroms = monomer length
<<L, the molecule is flexible at large scales
Walk is N steps : starts at , ends at . r = <,> the end-to-end vector.
R
N
denotes the sum over the lattice points The end to end vector is the sum over N “jump vectors”
P(r,N ) = (r,N )/(R
Distribution function for r
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S= Entropy
W= number of states (configurations)
Accessible to thermodynamic system
With energy E
From statistical mechanics:
()!!
"
#
$$
%
&
'
!!
"
#
$$
%
&
=
'
2
2
2/3
2
2
3
exp
3
2
,
Na
r
Na
N
r
P
(
From Boltzmann:
S(r) = k B ln[P(r,N)]
This expression is very important in
understanding the physics of polymers,
because it allows us to write down the
configuration entropy S(r) of a polymer
chain as a function of its elongation:
The surprising and important results of more careful calculations is that the presence of these short-range correlations does not alter the basic random walk character of the polymer chain statistics; it simply leads to an altered effective step size.
A more accurate picture builds the chain by successive steps and requires that the valence angles are used. ie. trans or gauche
> = nm
even for m does not equal n.
is a decreasing function of the
nm
chemical interval [m-n] (magnitude) and
decays exponentially for large values.
Thus the correlations are of finite range.
Place g consecutive vectors b into a
subunit c i If g is much larger than the the
range of correlations then c nm the new
vectors will be uncorrelated.
The problem is then for N/g independent
variables: leads to Gaussian statistics.
Providing N/g is large: ideal chain
behaviour is realised.
Extension of a chain, f and -f , require the average elongation <r>f of the chain Providing f is small
= 0[ 1 + 5/2 ] <<1
= 4C.N R3/3M
(volume fraction)
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The numerical results yield
N-1
Where z’ is smaller then z as for the
z = 6 and z’ = 4.68.
depends on the dimensionality:
3D = 7/6 and 2D = 4/3
R = aN where = 3/5
Here is a universal exponent
= 3/5, 2D = 3/4 and 1D
3D
Barnes H.A.
xy xz yz
polymerisation
density of / a chains per unit area. The volume per chain is 2/. We can write the total energy as the sum of stretching
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The value of the brush height h that minimizes the total energy gives us:
h ~ [ b
(1-2)]1/3 N
This reveals that a chain end-grafted to a polymer surface at high enough densities to overlap significantly with its neighbours is strongly stretched – the chain dimensions vary not as N 3/5, as
They do for an isolated chain in a good solvent, but as a linear function of N .。