for all N monomers in the chain this energy is N times larger.
F
≈ k
T
υ
N 2
(42)
B
R3
int
The Flory estimate for the entropic contribution to the free energy is the energy required to stretch an ideal chain to the end to end distance R.
F
≈ k
T
R2
(43)
NA2
ent
B
The total free energy is obtained by the sum of Eq.[42] and Eq.[43].
N
2
R
2
F = Fint + Fent
kBT υ
+
(44)
R
3
NA
2
The minimum free energy of the chain gives the equilibrium size RF. Taking ∂∂FR = 0 we obtain
∂F
2
RF
= 0 = kBT −3υ
N
+ 2
(45)
∂R
4
NA
2
RF
And finally
RF = υ15 A2 5 N 35
(46)
Thus, the size of a real chain is much larger than that of an ideal chain with the same number
of
monomers.
This
can
be expressed in terms of the swelling ratio
R
υ
N
1
15
for
υ
N
1
> 1. If the total interaction energy of a chain is smaller than
F
≈
2
2
1
A
3
A
3
AN
2
kBT then a chain will not swell.
The Flory theory thus leads to a universal power law
RF ~ N v ; v = 0.6
(47)
This theoretical result agrees very well with experimental results. Such a swollen chain has a fractal dimension df = 1/v that is df = 1.67 for swollen chains. However, the success of the Flory theory is due to a fortuitous cancellation of errors. The repulsion energy is overestimated because the correlations between monomers along the chain are omitted.
Flexible Polymers
13.29
Similarly the elastic energy is also overestimated because the ideal chain conformational entropy is assumed.
The correct statistical theory relates to the renormalization group which was introduced by Robert Wilson, in order to describe phase transitions, where universal behaviour is observed for T close to the critical temperature Tc as expressed by ε = (T - Tc)/Tc. For small ε universal behaviour takes place that is described by this renormalization group. De Gennes realized that for long chain polymers 1/N may be identified with ε and then the mathematics of the renormalization group can be taken over to describe the polymer conformations of real polymers. The correct renormalization group result for the swelling index is ν = 0.588 very close to the Flory result of ν = 0.6. This exponent also describes the statistics of self avoiding random walks.
5 Summary
In this lecture we have given an introduction into flexible polymers. First we had a look on the range of applications and the history of the development of polymers. Then we emphasized a number of distinguished contributors to polymer science. Thereafter, the basic definitions and properties of polymers were elucidated. In particular the importance of the molecular weight for chain properties was discussed and different notions for the molecular weight distributions were presented. In the second part of the lecture the chain conformations were discussed in some detail. We introduced the mean square end to end distance in terms of the characteristic ratio C∞ and defined the Kuhn segment that gives the equivalent freely jointed chain. We then considered probability distributions for any distance rij along the chain which came out to be Gaussian for chains without interactions between the monomers. For such a chain the free energy is purely entropic and leads to a Hookian relationship between force and end to end vector. Turning to scattering we defined the pair correlation function and derived the polymer form factor in terms of the Debye function. Finally we addressed in a simple approximation real chains, where interactions between monomers take place. This interaction is described in terms of an excluded volume that a given monomer cannot pervade if another monomer is present. Using the Flory approach we minimized the free energy. This leads to a new scaling relation for the end to end distance that follows a behaviour RE ≈ Nν with ν = 0.6. The scaling exponent turns out to be very close to the result of sophisticated renormalization group approaches that give ν = 0.588.
References
Text books further reading
P.J. Flory, Statistical Mechanics of Chain Molecules (Interscience Publishers, New York, 1969)
H.G. Elias, Makromoleküle (Hüthig + Wepf, Basel, Heidelberg, New York, 1990)
13.30
D. Richter
M. Rubinstein, R. Colby, Polymer Physics (Oxford University Press, 2003)
P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979)
J. des Cloizeaux, G. Jannik, Polymers in Solution: Their Modelling and Structure
(Clarendon Press, Oxford, 1990)
Flexible Polymers
13.31
Exercises
A E13.1
Silly Putty® is a typical viscoelastic material made from polydimethylsiloxane, silica and oils.
Propose and justify the 3 functions for the following behavior of Silly Putty. -When left on the table for a long period of time the height, h, decays in the manner of figure "a".
-When rapidly pulled and observed over only shot times the length, l, follows "b". -When pulled at intermediate speed (1) and then released (2) the behavior of "c" results.
B E13.2
The Rouse model is used to construct more complicated models for polymer dynamics that deal with chain entanglements such as the tube (reptation) model. In the tube model the polymer chain retains Rouse dynamics within the confinement of a rigid tube of cross-
sectional area <d2>, where d is the tube diameter. For simplicity we take d 2 = A2 , where A is the size of a Rouse unit. The tube follows a random walk in 3-D space.
a) If the tube length is Lp = NRA, where NR is the number of Rouse units in a chain, and if
the chain in the axial direction of the tube follows Rouse dynamics, with a friction factor of ξR for each Rouse unit,
-Whatis the total chain friction factor along the tube axis for the chain in the tube, ξp ? -How does this compare with the Rouse chain friction factor in 3D space?
13.32
D. Richter
b)The Einstein relationship tells us that the diffusion coefficent should follow D = kT/ξ. -What is the diffusion coefficient for the entire chain for motions along the length of the tube (along the tube axis)?
-What is the predicted molecular weight dependence of this diffusion coefficient? -Howdoes this compare with the Rouse diffusion coefficient?
c)For Brownian motion, such as motion of the chain in the tube, the distance traveled, d, in
time, t, is given by s = Dt .
-What is the average time for the chain to move the length of the tube (this is called the reptation time,τd )?
-Howdoes τd scale with molecular weight?
-Howdoes this compare with the scaling of the Rouse time, τR , with
molecular weight?
-Howdoes this compare with the observed relaxation time for entangled systems (same as the scaling behavior of η0 with molecular weight)?
d) The diffusion coefficient for centre of mass motion of a chain in a tube in 3D-space is determined by considering the size of the random walk tube in 3-d space, Re = NR1/2A, and the
time required to move the length of the tube. (D = (distance)2/time)
-What is the molecular weight dependence of the diffusion coefficient for chain motion in 3D-space for a chain confined to a tube?
-Howdoes this compare with the Rouse diffusion coefficient you gave above?
14Applications of Neutron Scattering - an Overview
Lecture Notes of the JCNS Laboratory Course Neutron Scattering (Forschungszentrum Jülich, 2011, all rights reserved)
14.2
Th. Brückel
14.1 Introduction
Fig. 14.1: Lengthand time scales covered by research with neutrons giving examples for applications and neutron techniques [1].
Research with neutrons covers an extraordinary range of lengthand time scales as depicted in figure 14.1. The very extremes of length scales - below 10-12m - are the domain of nuclear and particle physics, where e. g. measurements of the charge or electric dipole moment of the neutron provide stringent tests of the standard model of particle physics without the need of huge and costly accelerators. On the other extreme, neutrons also provide information on lengthand time scales relevant for astronomical dimensions, e. g. the decay series of radioactive isotopes produced by neutron bombardment give information on the creation of elements in the early universe. In this course, however, we are only concerned with neutrons as a probe for condensed matter research and therefore restrict ourselves to a discussion of neutron scattering. Still, the various neutron scattering techniques cover an area in phase space from picometers pm up to meters and femtoseconds fs up to hours, a range, which probably no other probe can cover to such an extend.
Different specialized neutron scattering techniques are required to obtain structural information on different length scales:
With wide angle neutron diffractometry, magnetization densities can be determined within single atoms on a length scale of ca. 10 pm1. The position of at-
1 In this sense, neutrons are not only nanometer nm, but even picometer pm probes!
Applications neutron scattering
14.3
oms can be determined on a similar length scale, while distances between atoms lie in the 0.1 nm range2.
The sizes of large macromolecules, magnetic domains or biological cells lie in the range of nm to μm or even mm. For such studies of large scale structures, one applies reflectometry or small angle scattering techniques.
Most materials relevant for engineering or geo-science occur neither in form of single crystals, nor in form of fine powders. Instead they have a grainy structure, often with preferred orientation of the grains. This so called texture determines the macroscopic strength of the material along different directions. Texture diffractometry as a specialized technique allows one to determine this grainy structure on length scales of up to mm.
Finally, for even larger structures, one uses imaging techniques, such as neutron radiography or tomography, which give a two dimensional projection or full 3- dimensional view into the interior of a sample due to the attenuation of the neutron beam, the phase shift or other contrast mechanisms.
In a similar way, different specialized neutron scattering techniques are required to obtain information on the system’s dynamics on different time scales:
Neutron Compton scattering, where a high energy neutron in the eV energy range makes a deep inelastic collision with a nucleus in so-called impulse approximation, gives us the momentum distribution of the atoms within the solid. Interaction times are in the femtosecond fs time range.
In magnetic metals, there exist single particle magnetic excitations, so-called Stoner excitations, which can be observed with inelastic scattering of high energy neutrons using the so-called time-of-flight spectroscopy or the triple axis spectroscopy technique. Typically, these processes range from fs to several hundred fs.
Lattice vibrations (phonons) or spin waves in magnetic systems (magnons) have frequencies corresponding to periods in the picosecond ps time range. Again these excitations can be observed with time-of-flightor triple axis spectroscopy.
Slower processes in condensed matter are the tunneling of atoms, for example in molecular crystals or the slow dynamics of macromolecules. Characteristic time scales for these processes lie in the nanosecond ns time range. They can be observed with specialized techniques such as backscattering spectroscopy or spinecho spectroscopy.
Even slower processes occur in condensed matter on an ever increasing range of lengths scales. One example is the growth of domains in magnetic systems, where domain walls are pinned by impurities. These processes may occur with typical time constants of microseconds μs. Periodic processes on such time scales can be observed with stroboscopic neutron scattering techniques.
Finally, kinematic neutron scattering or imaging techniques, where data is taken in consecutive time slots, allow one to observe processes from the millisecond ms to the hour h range.
2 In what follows, we use as “natural atomic unit” the Ångstrøm, with 1 Å=0.1 nm.
14.4
Th. Brückel
In this chapter, we will overview the various techniques used in neutron scattering and provide some examples for their application. We will start by repeating the properties of the different correlation functions, in order to be able to judge what kind of information we can obtain from a certain neutron scattering experiment. We will introduce neutron scattering techniques used to obtain information on “where the atoms are” (diffractometry) and “what the atoms do” (spectroscopy). We will finish by reviewing the range of applicability of various neutron scattering methods and compare them to other experimental techniques.
14.2 Scattering and correlation functions
This somewhat advanced section can be skipped during first reading, but is given here for completeness.
The neutron scattering cross section for nuclear scattering can be expressed in the following form (for simplicity, we restrict ourselves to a mono-atomic system):
2
k '
N
Sinc (Q, ) |
|2
(Q, )
(14.1)
| b |2
|
|2
Scoh
b
b
k
The cross section is proportional to the number N of atoms. It contains a kinematical factor k’/k, i. e. the magnitude of the final wave vector versus the magnitude of the incident wave vector, which results from the phase-space density. The scattering cross section contains two summands: one is the coherent scattering cross section, which de-
pends on the magnitude square of the average scattering length density | b |2 and the other one is the incoherent scattering, which depends on the variance of the scattering length | b |2 | b |2 . The cross section (14.1) has a very convenient form: it separates
the interaction strength between probe (the neutrons) and sample from the properties of the system studied. The latter is given by the so-called scattering functions Scoh (Q, )
and Sinc (Q, ) , which are completely independent of the probe and a pure property of
the system under investigation [2]. The coherent scattering function Scoh (Q, ) (also
called dynamical structure factor or scattering law) is a Fourier transform in space and time of the pair correlation function:
Scoh
(Q, )
1
G(r,t)ei(
Q
r t)d3rdt
(14.2)
2
Here the pair correlation function G(r,t) depends on the time dependent positions of the atoms in the sample:
G(r,t) N1 (r ' ri (0)) (r ' r rj (t)) d3r '
ij
(14.3)
N1 (r ',0) (r ' r,t)d3r '
Applications neutron scattering
14.5
ri (0) denotes the position of atom i at time 0, while r j (t) denotes the position of an-
other atom j at time t. The angle brackets denote the thermodynamic ensemble average, the integral extends over the entire sample volume and the sum runs over all atom pairs in the sample. Instead of correlating the positions of two point-like scatterers at different times, one can rewrite the pair correlation function in terms of the particle density as given in the second line of (14.3). Coherent scattering arises from the superposition of the amplitudes of waves scattered from one particle at time 0 and a second particle at time t, averaged over the entire sample volume and the thermodynamic state of the sample. In contrast, incoherent scattering arises from the superposition of waves scattered from the same particle at different times. Therefore the incoherent scattering function Sinc (Q, ) is given in the following form:
Sinc (Q, )
1
Gs (r,t)ei(
Q
r t)d3rdt
(14.4)
2
of the self
correlation function
which is the Fourier transform in
space and time
GS (r,t):
G (r,t)
1
(r
' r (0)) (r ' r r (t)) d3r '
(14.5)
N
s
j
j
j
We next define the intermediate scattering function S(Q,t) as the purely spatial Fourier
transform of the correlation function (here we have dropped the index “coh” and “inc”, respectively, as the intermediate scattering function can be defined for coherent as well as for incoherent scattering in the same way):
S(Q,t): G(r,t)eiQ r d3r
(14.6)
S(Q, ) S '(Q,t)
For reasons, which will become apparent below, we have separated in the second line the intermediate scattering function for infinite time
S(Q, ) lim S(Q,t)
(14.7)
t
from the time development at intermediate times. Given this form of the intermediate scattering function S(Q,t) , we can now calculate the scattering function as the temporal
Fourier transform of the intermediate scattering function:
d 2 
= A2 , where A is the size of a Rouse unit. The tube follows a random walk in 3-D space.
(
