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C 1 Flexible Polymers

H. Frielinghaus

Institut fur Festkorperforschung

Forschungszentrum Julich GmbH

Contents

1 Introduction 2

2 Chain Models 22.1 The Freely Jointed Chain . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 32.2 The Freely Rotating Chain . . . . . . . . . . . . . . . . . . . . . . . . . .. . 62.3 The Kuhn Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Semiflexible Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92.5 Summary of Chain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .112.6 Distribution of the End-to-End Distance . . . . . . . . . . . . .. . . . . . . . 122.7 The Gaussian Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142.8 The Continuous Description . . . . . . . . . . . . . . . . . . . . . . . .. . . 15

3 Phenomena of Polymers 173.1 Polymer near a hard wall . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 173.2 Rubber elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 193.3 Scattering of chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 223.4 Deviations – The Flory Exponent . . . . . . . . . . . . . . . . . . . . .. . . . 243.5 Semidilute Solutions – The Blob Concept . . . . . . . . . . . . . .. . . . . . 27

4 Conclusion 29

A Appendix: Theorem of Lagrange 31

C1.2 H. Frielinghaus

1 Introduction

Polymers occur everywhere – in nature and industrial products. Examples are the DNA, wood,silk, synthetic polymers, often called plastics, fibres andgels. In biology the conformationof many molecules plays an important role, say of proteins orthe DNA, which assigns to themolecule a specific function. Arrangements of proteins may appear as a polymeric structureon a larger scale, again for specific tasks. For technological applications the easy tailoringof polymeric materials is important. Finally, technical/synthetic plastics are irreplaceable ineveryday life. Examples are housing of electronics, wire isolation, glue, rubber or medicalmaterials. Some of them are evidence of high-tech products.Many synthetic polymers are rather simple. They consist of asingle unit which is chemicallylined up (or chained) to form a linear polymer. One example ofsuch a unit is the monomerethylene (Fig. 1). The double bond gives rise to chemical reactions with other molecules of thesame species. If only two monomers have reacted a dimer is formed. If the reaction can be heldup, a polymerization process takes place, and finally a long chain molecule is formed. Whereasthe molecules look straight on the paper, they are not in reality, however. Many conformationsare possible, which will be the subject of this lecture.Additionally, polymers have an unconventional phase diagram. There is no gas phase. The “liq-uid” phase appears to be highly viscous. But besides the viscous contributions there are alsoelastic properties. These become especially obvious for rubber. Polymers also never crystallizecompletely. For many polymers the glassy state seems to be preferred. Besides the neat poly-mers, solutions of polymers also show a rich variety of properties. The large number of degreesof freedom per molecule is often responsible for the unfailing nature of polymers.Polymers in Biology differ from synthetic plastics. Often there is a larger variety of monomersbuilding up the chain molecule (DNA for example). The chain structure usually becomes visi-ble on larger scales – for DNA beyond the helix structure. Proteins (and DNA) are complicateddue to the folding, which leads to different hierarchial stages. Some of these stages might ap-pear as simpler polymeric structures (chromatin loops). Finally, special proteins make up thecytoskeleton by lining up to long filaments. The diameters ofsuch filaments lie in the rangeof 7-25nm. These filaments are responsible for cell stability, motion, and transport of materialwithin the cell (see contribution of A. Bausch C.9). In the living cell the polymerization anddepolymerization can be a continuous process. Through stabilization mechanisms the struc-tures are highly resistent. In many cases the properties of filaments on larger length scales canbe described by two parameters – the elasticity and the contour length. The coarse grainingaverages over the elementary properties of the microscopicpattern.In chapter 2 different chain models are developed to describe conformations of polymers. Thefirst model is rather simple and refinements allow for more realistic modelling. Chapter 3describes more phenomena of polymers where the developed concepts of microscopic confor-mations are used. The underlying theories and concepts are acollection of different textbooks[1–9].

2 Chain Models

The simplest synthetic polymer is polyethylene. While we already described the polymerizationprocess, we now want to consider the conformation of such a chain. The main task will beto describe the position of the carbon atoms – or more generalized the arrangement of the

Flexible Polymers C1.3

HH

HH

H

HH

HH

H

HH

HH

HH

HH

HH

HH

HH

HH

HH

HH

H

HH

HH

HH

HH

HH

HH

H

HH

HH

Fig. 1: The steps from a monomer to a polymer. (top left) the monomer ethylene – actually flat,(top right) the saturated dimer, (bottom) the polymer polyethylene.

Fig. 2: The polyethylene chain with definitions of angles:θi bond angle,ϕi torsional angle.

polymer backbone. Each backbone atom is numbered by the index i. There already exist somerestrictions due to the bond anglesθi and the torsional anglesϕi. During the refinement of themodels the properties become more and more realistic. Finally, the positions of the hydrogenatoms – or of larger side groups – give rise to steric hindrance interactions and are consideredin the section of semiflexible polymers.

2.1 The Freely Jointed Chain

This particular chain is described by N consecutive backbone bonds, which do not have anyrestrictions [1, 10]. This means that the bond length is fixed(ℓ), but the angle between consecu-tive bonds is not restricted at all – restrictions due to the bond angles are neglected. The path ofthis backbone resembles that of a random walk. A drunken man on his way home would followa similar path since he does not remember which way he took a minute ago. Therefore, there isan obvious link to diffusion of particles already now, but wewill discuss this later (see section2.8).One observable of a chain is the end-to-end distance. It simply can be described by the sum ofall bond vectors (see Fig. 3) connecting two consecutive monomers (~ri ≡ ~ri−1,i; the generalized


Fig. 3: A sketch of a freely jointed chain. The bond vector has a constant lengthℓ, but the angleis totally free. The end-to-end distance~Ree is described by the sum of all bond vectors.

vector~ri,j connects two monomersi andj, which is shown in Fig. 4)

~Ree =N∑

i=1

~ri (1)

Statistically, the average of the end-to-end distance is simply zero. This is true since the freechain does not have a preferential orientation, and this means that there is no orientation of anyvector. This applies for every single bond vector and the end-to-end vector.

⟨

~Ree

⟩

= ~0 (2)

The equation above describes the first moment of the end-to-end distance distribution (see alsosection 2.6). The next moment is the second moment〈~R2

ee〉. But before averaging the end-to-end distance, some rearrangements of the terms should be made. The square of the end-to-enddistance is a scalar, and a square of a sum contains all possible pairs of products.

~R2ee = ~Ree · ~Ree =

N∑

i,j=1

~ri · ~rj (3)

Those pairs can be sorted for diagonal terms (r2i ) and off-diagonal terms, where the indexing

even can be sorted.

~R2ee =

N∑

i=1

~r 2i + 2 ·

∑

1≤i<j≤N

~ri · ~rj (4)


(i)

(i+1)

( j)

( j−1)ri,j

center ofmass

sj

si

Fig. 4: The definition of the vectors needed in the radius of gyrationcalculation. The vectors~si

describe the position of the monomeri relatively to the center of mass. The vectors~ri,j connectthe monomersi andj, and therefore describe a sub-chain.

The averaging of these two different sums yields a simple result: The diagonal terms considerthe squared length of a single bond (ℓ2), the summation of which simply adds upN times thesame number. The other contribution considers the correlation between different bonds. Oursimple model neglects any correlations for now and the term vanishes.

⟨

~R2ee

⟩

=N∑

i=1

⟨~r 2

i

⟩+ 2 · 0 = Nℓ2 (5)

The characteristic size of the polymer chain is given by the square root of the second moment.There is no preferential orientation – but the polymer needssome space.

√⟨

~R2ee

⟩

=√

N ℓ (6)

The for polymers characteristic dependence of the average end-to-end distance〈~R2ee〉1/2 on the

degree of polymerizationN occurs already for this over-simplified model. The average chainsize is proportional to

√N only and therefore is much smaller than the full extension. This is

due to the large number of degrees of freedom of such a molecule.Another frequently used observable is the radius of gyration. It measures the size of a moleculein the manner of a moment of inertia. The vectors of all monomers relatively to the centerof mass are squared (assuming the same weight of all monomers, see Fig. 4). This sum isnormalized by the number of monomers (N +1). The radius of gyration therefore measures theaverage extension of a chain relatively to the center of mass.

~R2g =

1

N + 1

N∑

i=0

~s 2i (7)

The Theorem of Lagrange allows for the rearrangement of thissum (Appendix A). Now allvectors connecting monomersi andj occur. Note that the normalization is(N + 1)2 due to the


larger amount of addends now.

~R2g =

1

(N + 1)2

∑

0≤i<j≤N

~r 2i,j (8)

This would not help much, unless the connecting vectors can be interpreted as end-to-end vec-tors of chain parts. The single connecting vector is a sum of bond vectors

~ri,j =

j∑

k=i+1

~rk (9)

and in the averaging (of the radius of gyration) the square ofthe connecting vector yields simply(j − i)ℓ2.

⟨

~R2g

⟩

=1

(N + 1)2

∑

0≤i<j≤N

(j − i) ℓ2 (10)

=ℓ2

(N + 1)2

N∑

j=1

j∑

k=1

k (11)

=ℓ2

(N + 1)2

N∑

j=1

j · (j + 1)

2(12)

=N · ℓ2

6· N + 2

N + 1−→N≫1

N · ℓ2

6(13)

The term(j − i) describes a length of a sub-chain, and all possible combinations have to besummed up. This summation can be sorted by summing only termswithin a chain of lengthj(instead ofN). In this chain all possible lengthsk of sub-chains are summed up. The inner sumis an arithmetic sum. The outer sum can be determined by mathematical tables or algebraicprograms (Maple, Mathematica, ...). In the limit of long chains the second factor of eq. 13reduces to unity. Therefore there is a fixed ratio between thesquares of the radius of gyrationand the end-to-end distance.

⟨

~R2g

⟩

=

⟨

~R2ee

⟩

6(14)

The radius of gyration is smaller since it measures relatively to the center of mass, whereas theend-to-end distance is a measure for the whole chain.In the calculation of the radius of gyration we used the argument that sub-chains behave similarto the whole chain (see eq. 9), except that the chain length isjust (j − i). This means that thechain has some self-similarity when looking inside the chain. Scattering experiments (sections3.3 and 3.4) will display this behavior. A chain is a fractal object therefore.

2.2 The Freely Rotating Chain

The freely rotating chain model accounts for the fact that consecutive carbon-carbon-bondsenclose a bond-angleθ typical for a tetrahedron (see Fig. 2). This is due to the sp3-hybridizationof the carbon atom. The torsional angleϕ is still assumed to be free. The effect of steric


Fig. 5: A sketch of a freely rotating chain. The bond angleθ is fixed, whereas the torsionalangle is free.

hindrance of hydrogens or even side groups will be discussedlater. The resulting chain modelis depicted in Fig. 5. It is directly obvious that the projection of two successive bonds is:

〈~ri · ~ri+1〉 = ℓ2 cos θ (15)

In detailed calculations [10] it can be shown that the correlation between bond vectors of largerdistances decays in the following way:

〈~ri · ~rj〉 = ℓ2(cos θ)|j−i| (16)

For the already derived formula of the end-to-end distance (compare eq. 4) the second addendhas to be taken into account now and is finite.

⟨

~R2ee

⟩

=

N∑

i=1

⟨~r 2

i

⟩+ 2 ·

∑

1≤i<j≤N

〈~ri · ~rj〉 (17)

= Nℓ2 + 2ℓ2∑

1≤i<j≤N

(cos θ)j−i (18)

= Nℓ2 + 2ℓ2

N−1∑

k=1

(cos θ)k(N − k) (19)

The decay of correlations can be directly inserted and the power series can directly be executed.The numberk tells us how long a considered sub-chain is, and the factor(N − k) accountsfor the number of possibilities where a sub-chain has the first segment. The remaining sumis a modified power series and again can be determined by mathematical tables or algebraicprograms.


⟨

~R2ee

⟩

= Nℓ2

[1 + cos θ

1 − cos θ− 2 cos θ

N· 1 − (cos θ)N

(1 − cos θ)2

]

(20)

−→N≫1

Nℓ2

[1 + cos θ

1 − cos θ

]

(21)

While the full result looks overwhelming, we first consider the case of long chains. Then thesecond term of eq. 20 can be neglected. Later we will also consider shorter chains (section2.4) where the lower flexibility plays an important role. Comparing eq. 21 with 5 we see, thatthe principal dependence of the end-to-end distance on the bond numberN is conserved. Thecharacteristic ratio of the freely rotating chain is definedfor large bond numbers relatively tothe freely jointed chain:

C∞ = limN→∞

⟨

~R2ee

⟩

/(Nℓ2) (22)

And for the case of a freely rotating chain we simply obtain:

C∞ =1 + cos θ

1 − cos θ(23)

For tetrahedral bond angles (θ = 70.53◦) the characteristic ratio isC∞ = 2. This is still anidealized result. Polymeric sulphur might come close to theidealization. For real polymers thecharacteristic ratio can be defined by experiments, and one typically obtains values forC∞ inthe range of 4 to 10 [11, 12]. Larger values correspond to a higher stiffness. The reason forthis result is a steric interaction of hydrogens in the case of polyethylene or even larger sidegroups. Models exist which take this steric hindrance into account and obtain quite good resultsfor the experimentally observedC∞ values [1, 10]. For our purpose we can argue that the sterichindrance leads to a higher stiffness, which leads to a slower decay of the correlations thanthe freely jointed chain (eq. 16). The argumentcos θ has to be substituted then. In section2.4 we will introduce a rigidity as a coarse grained parameter to describe the slower decay ofcorrelations. Note that the characteristic ratioC∞ bases on the microscopic bond with a typicallength of 1.54A andN counts the number of chemical bonds in the molecule.

2.3 The Kuhn Chain

We have argued that the correlations between bonds of longerdistances|j − i| decay in themanner of equation 16. This means that sub-chains can define an effective segment (the Kuhnsegment, see Fig. 6) with negligible correlations.

⟨

~R2ee

⟩

K=

NK∑

i=1

⟨~r 2

i

⟩

K+ 2 ·

∑

1≤i<j≤NK

〈~ri · ~rj〉K (24)

= NKℓ2K (25)

The segment length of the Kuhn chain is not a constant anymore, but the average length can begiven byℓK . The Kuhn bond number is calledNK . Now, to relate the new effective parametersℓK andNK with microscopic models we need to find the model-independent observables. Oneof them is the contour length (given for either model):


Fig. 6: The Kuhn chain embraces several segments to form a single effective segment. Thiseffective segment has lost all correlations with respect tothe others.

L = NKℓK = Nℓ (26)

The other one is the end-to-end distance:⟨

~R2ee

⟩

K= LℓK = C∞Lℓ (27)

From the model-independent observables one directly can solve for the Kuhn chain definingparameters, and one obtains:

ℓK = C∞ℓ and NK = N/C∞ (28)

In literature [2] there exist also other expressions.1 Here, the basis model is the freely ro-tating chain, and the full extension of this chain model is taken as the model-independentobservable. The slightly corrected Kuhn chain parameters read thenℓK =

√1.5C∞ℓ and

NK = N/(1.5 C∞). This example demonstrates that the Kuhn model depends on the micro-scopic definition of a contour length. Whenever possible, theoreticians like to take the end-to-end distance as the only model-independent observable. On the other hand, the Kuhn segmentlength provides a good estimate over which distance correlations between bond vectors are lost.We will therefore use the Kuhn length in the following chapters, and some results actually willbe independent on the definition of a contour length (see section 3.1).

2.4 Semiflexible Chains

Real chains have finite length and the result of eq. 20 has to bediscussed in detail. The men-tioned result bases on a microscopic model (freely rotatingchain). Another model [13] consid-ers the chain in a coarse grained manner. The decay of correlation of tangential vectors over acontour length segmentℓ is then described by:

1There exist even more effective segment lengths, which leave the number of segmentsN constant instead ofthe contour length. Those models just compare a freely rotating chain with a freely jointed chain with identicalN .Scattering experiments have access to this number as we willsee in section 3.3 (especially eq. 83).


〈cos θ〉 = exp(−α) (29)

The parameterα is then given by the coarse grained rigidityκ:

α = ℓkBT

κ(30)

Even though the coarse grained model allows to calculate theend-to-end distance in a consistentway, we just substitute thecos θ terms and obtain for smallα (the productαN is not necessarilysmall):

⟨

~R2ee

⟩

= Nℓ2

[1 + exp(−α)

1 − exp(−α)− 2 exp(−α)

N· 1 − exp(−αN)

(1 − exp(−α))2

]

(31)

−→α≪1

Nℓ2

[2

α− 2

N· 1 − exp(−αN)

α2

]

(32)

= 2[Npℓ

2p − ℓ2

p(1 − exp(−Np))]

(33)

The last line involves the persistence lengthℓp and the corresponding degree of polymerizationNp. The definitions are:

ℓp = ℓ/α =κ

kBTand Np = αN (34)

Depending on the rigidityκ, the parameterα can be large or small, and correspondinglyNp islarge or small. The limits correspond to a flexible chain or a rod, and the end-to-end distancebecomes for these limits:

⟨

~R2ee

⟩

−→Np≫1

2Npℓ2p and

⟨

~R2ee

⟩

−→Np≪1

2N2p ℓ2

p (35)

The flexible chain limit has the typical proportionality toNp and ℓ2p. The rod-like limit is

proportional to the squared contour length∝ (Npℓp)2. Examples in biology are found for

different filaments of the cytoskeleton or the neuronal network (see Fig. 7 and reference [14]).Microtubule are rather stiff, and thereforeNp is rather small. The considered actin filaments areof intermediate stiffness, whereas neurofilaments appear to be highly flexible, which implies alargeNp. Here, the coarse graining means that the properties of the microscopic filament unitsappear only in the manner of an averaged flexibility.Another option to express the behaviour of a semiflexible chain is using the Kuhn length anddegree of polymerization. Since we declared the Kuhn model to be the most useful we willcontinue in this notation. The end-to-end distance then is:

⟨

~R2ee

⟩

= NKℓ2K − 1

2ℓ2K(1 − exp(−2NK)) (36)

The basic units are connected by:

ℓK = 2ℓp and NK = 12Np (37)

The limits for flexible chains and rigid rods read then:⟨

~R2ee

⟩

−→NK≫1

NKℓ2K and

⟨

~R2ee

⟩

−→NK≪1

N2Kℓ2

K (38)


Fig. 7: Biological examples of semiflexible chains [14]: Microtubule are rather stiff; the F-actin is semiflexible; the neurofilament is highly flexible. The number of persistence lengthsNp

represents a measure of the flexibility.

In the same manner as eqs. 31-33 one can derive an expression for the radius of gyration. Butalso for a continuous contour length [15] (and the Wiener integral method [16]) one obtains theidentical result. We will just report the result which reads:

⟨

~R2g

⟩

=NKℓ2

K

6− ℓ2

K

4+

ℓ2K

4NK− ℓ2

K

8N2K

[1 − exp(−2NK)] (39)

The limits of flexible chains and rigid rods for the radius of gyration read then:⟨

~R2g

⟩

−→NK≫1

16NKℓ2

K and⟨

~R2g

⟩

−→NK≪1

112

N2Kℓ2

K (40)

A full comparison of the end-to-end distance and the radius of gyration is depicted in Figure 8.In the limit of large degrees of polymerization one finds the flexible chain limit with a ratio of6 between the two radiiRg andRee. In the limit of small degrees of polymerization one findsthe rigid rod limit with a ratio of 12 between the two radii.

2.5 Summary of Chain Models

For all chain models we can summarize the characteristic behavior of flexible chains where thecharacteristic size increases with the square root of the degree of polymerizationN .

√⟨

~R2ee

⟩

∝√⟨

~R2g

⟩

∝√

N · ℓ (41)

This indicates that the characteristic size is much smallerthan the full extension. The explicitprefactors depend on the flexibility of the chain and can be expressed in terms of a microscopic


10-2 10-1 10010-3

10-2

10-1

100

6

12

Rg

rigid rod

rigid rod

flexib

le chain

Ree

flexib

le chain

1/NK

Rg a

nd

Ree i

n u

nits

(NK

2 "K

2 )

Fig. 8: The comparison of the two typical radiiRg andRee as a function of the number of Kuhnsegments. On the left (largeNK) the flexible chain limit is reached. On the right (smallNK)the rigid rod limit is reached.

modelC∞ (eq. 21), a coarse grained model (eq. 35) with a rigidityκ, or more sophisticatedmicroscopic models [1, 10]. In the rigid rod limit, the characteristic sizes are proportional tothe full extension of the chain:

√⟨

~R2ee

⟩

∝√⟨

~R2g

⟩

∝ N · ℓ (42)

For this limit the flexibility does not play any role, andRee measures exactly the full extension(see eq. 26 and discussion below).

2.6 Distribution of the End-to-End Distance

Previously we have calculated averages of the characteristic radii. In this section we want toderive an expression for the distribution of the end-to-enddistance [3]. Without many restric-tions, we want to start with (a) either the freely rotating chain or (b) the Kuhn chain. This meanswe want to use the statistical independence of the chain segments by (a) definition or (b) forlarge enough segments, which have lost correlations. For the individual distribution of a singlesegment the normalization holds:

∫

d3ri pi(~ri) = 1 (43)

Later we will see that the final result does not depend on the shape of the single segment distri-bution. The first moment expresses that there is no preferredorientation of the segments:

〈~ri〉 =

∫

d3ri ~ripi(~ri) = ~0 (44)


The second moment yields the average size of the single segment, as we have seen before.

⟨~r 2

i

⟩=

∫

d3ri (~r2i )pi(~ri) = ℓ2 (45)

These considerations focused on moments of a single segmentdistribution. The statistical in-dependence of the segments is expressed by:

〈~ri~rj〉 = δijℓ2 (46)

Now the observable of the end-to-end distance is introducedby ~R =∑

i ~ri. The distributionof this observable is simply expressed by the average under the restriction that the conditionbetween~R and

∑

i ~ri holds:

P (~R) =

⟨

δ(~R −∑

i

~ri)

⟩

(47)

Now one uses the trick of the Fourier transformation of this distribution. The advantage is, thatthe Fourier variable~Q directly couples with each individual segment variable independently.

P ( ~Q) =

⟨∫

d3R exp(i ~Q~R) · δ(~R −∑

i

~ri)

⟩

(48)

=

⟨

exp(i ~Q∑

i

~ri)

⟩

(49)

i is the imaginary unit. The averaging of this expression is now written explicitly, and transfor-mations of the exponential are done consequently. The higher order terms must be taken intoaccount as well and will be discussed below.

P ( ~Q) =

∫

d3r1 p1(~r1) · · ·∫

d3rN pN(~rN) exp(i ~Q∑

i

~ri) (50)

=

∫

d3r1 p1(~r1) · · ·∫

d3rN pN(~rN)

[

1 − 12( ~Q∑

i

~ri)2 + · · ·

]

(51)

=

∫

d3r1 p1(~r1) · · ·∫

d3rN pN(~rN)

[

1 − 12

∑

i

( ~Q~ri)2 + · · ·

]

(52)

The first transformation just involves the Taylor series of the exponential. The first momentsof ~ri vanish since there is no preferential orientation. The nextline uses the statistical inde-pendence of the chain segments. In the following we do not usethe explicit formulation of theaveraging, but use the brackets again:

P ( ~Q) = 1 − 16

∑

i

⟨

~Q2~r 2i

⟩

+ · · · (53)

= 1 − 16N ~Q2ℓ2 + · · · (54)

≈ exp(−16N ~Q2ℓ2) (55)


Y

X

Y

XFig. 9: The distribution of a single segment (freely jointed chain)and the whole chain. Whilefor a single segment the distribution is rather pronounced,for a large amount of segments thecentral limit theorem yields a Gaussian distribution.

The first line is valid for squared averages of individually projected segments. Please note theadditional factor1

3. Now, the individual second moments occur, the segment length appears as

the explicit average. Finally, the first two terms of the Taylor series appear equally well for anexponential.Actually one had to take all higher order terms of eq. 51 into account to be sure that the finalexpression holds. In the higher order terms there occur someexpressions with higher ordermoments of single chain segments andmanyexpressions involve only second order momentsof different segments. This“many” becomes the more important the longer the chain is. Thesemanyterms finally make the exponential exact in the limit of long chains.Finally using the definition of the mean end-to-end distanceR2

ee = Nℓ2, and Fourier transform-ing the last expression back into real space, one obtains:

P (~R) = (23πR2

ee)−3/2 exp(−3

2

~R2

R2ee

) (56)

This distribution is a Gaussian with the respective normalization, which is independent of theinitial distribution. The justification of eq. 55 might seemto be hand-waving; the final result isexact in the context of the central limit theorem [17].While other explicit derivations for the freely rotating chain exist (see “saddle point approxi-mation” [10]), we explicitly want to demonstrate the meaning of the central limit theorem. Forthe freely rotating chain the distribution of a single segment covers a sphere (see Fig. 9), whilefor the whole chain the end-to-end distance distribution isa Gaussian. So, even a pronounceddistribution of a single segment ends up in a Gaussian distribution for a whole chain, when alarge number of segments is considered.

2.7 The Gaussian Chain

The step to the Gaussian chain is small taking all previous sections into account [4]. The idea is,that the number of segments in a Kuhn segment is already largeenough, so that the distribution


Fig. 10: The equivalent picture for a Gaussian chain is a number of effective monomers con-nected by elastic springs.

function of a single Kuhn segment can be approximated by a Gaussian distribution. Then thedistribution function ofNK segments can be written:

W (~r1, · · · , ~rNK) = (2

3πℓ2

K)−3NK/2 · exp(− 3

2ℓ2K

∑

i

~r 2i ) (57)

The limit of a Gaussian distribution might be achieved for slightly larger segments. So thenumbersℓK andNK should be taken as approximate values, which they are anyway(see section2.3). Finally we write for the effective bond vectors the difference of the monomer vectors:

W (~R0, · · · , ~RNK) = (2

3πℓ2

K)−3NK/2 · exp

(

− 3

2ℓ2K

∑

i

(~Ri − ~Ri−1)2

)

(58)

We will need this expression in the next section in the mannerof an equivalence. Please notethat the normalization does not cover the initial vector~R0, which will be discussed in oneexample below (section 3.1). The now obtained distributionfunction can be compared with athermodynamic partition function by rewriting:

W ∝ exp(

− 1

kBT· 3kBT

2ℓ2K

∑

i

(~Ri − ~Ri−1)2

︸︷︷︸

Hamiltonian

)

(59)

kB is the Boltzmann constant. The identified Hamiltonian describesNK Hookean springs witha single temperature dependent spring constant. The Gaussian chain therefore can be drawnas beads connected byNK springs (see Fig. 10). The temperature dependence of the springconstant is important for the rubber elasticity as we will see in the example below (section3.2). Furthermore, the temperature dependent Hamiltoniancan be called entropic since thetemperature dependence actually cancels out.

2.8 The Continuous Description

While we had a certain number of beads in the previous chapter, we now want to considerthe limit of a continuous chain path in space. The sum in the exponential of eq. 58 is simplyreplaced by an integral and the difference of the monomer vectors is substituted by a differential,and so we obtain for the distribution function:


Table 1: Equivalence of the discrete and continuous description of aGaussian chain. Note thatthe continuous variablesk or s are still notated in units of indices.

Discrete Continuousk′∑

i=k

k′∫

k

ds

~Ri − ~Ri−1 ∂ ~R/∂s

~Ri+1 − 2~Ri + ~Ri−1 ∂2 ~R/∂s2

Kronecker deltaδij Dirac delta functionδ(s − s′)

integral∫

d3R1 · · ·∫

d3RNKpath integral

∫δ3R

W = exp

− 3

2ℓ2K

∫

ds

(

∂ ~R(s)

∂s

)2

(60)

Again, we can identify a Hamiltonian with a temperature dependent spring constant. Generally,one introduces a potential for the beadsUe.

H =3kBT

2ℓ2K

N∫

0

ds

(

∂ ~R

∂s

)2

+

N∫

0

ds · Ue(~R) (61)

The partition function or the distribution function is thenobtained in the manner of classicalthermodynamics:

G ∝ exp

(

− 1

kBTH

)

(62)

The change in the nomenclature going from the discrete to thecontinuous description of chainsis summarized in Table 1: The summation over monomers is substituted by an integration. Thedifference between neighbored monomer vectors is substituted by the differential. The givendifference between three neighbored monomer vectors is substituted by the second derivative.The discrete Kronecker delta is replaced by the Dirac delta function. In a partition function theintegration over the monomer vectors is replaced by a path integral. Taking all these rules intoaccount and using the correct normalization the simple equation 62 is more precisely given by:

G(~R, ~R′, N) =

~R′∫

~R

δ3R · exp

(

− 32ℓ2

K

N∫

0

ds(

∂ ~R∂s

)2

− 1kBT

N∫

0

ds · Ue(~R)

)

∫d3R′

~R′∫

~R

δ3R · exp

(

− 32ℓ2

K

N∫

0

ds(

∂ ~R∂s

)2) (63)

This Green function has the constraints of the initial vector (the first monomer vector) and thefinal vector (last monomer) and the degree of polymerization. This formalism is used to describethe thermodynamic behavior of polymers [4] partly in blends, partly in solutions as we will seein the next section. One important property of this distribution function is, that it is a Greenfunction, which solves the following differential equation:


N = 12

N = 3N = 00

R'

Fig. 11: Comparison of the Green function of polymers with the diffusion process: (left) discretepicture, (right) continuous picture.

(∂

∂N− ℓ2

K

6∇2

~R+

1

kBTUe(~R)

)

G(~R, ~R′, N) = δ(~R − ~R′)δ(N) (64)

The left hand side term is the homogenous part and the right hand side is the inhomogeneitytypical for Green functions. The solutions forN > 0 solve the homogenous part alone, andsome interpretations of the left terms can be given: First ofall we are talking aboutpolymerthermodynamics. The first two addends are typical for adiffusion equationwhereN corre-sponds to time. And finally the whole left part looks like aSchrodinger equation, but the time isimaginary (N). A comparison of the polymer thermodynamics and the diffusion interpretationis given in Figure 11. The first step (N = 0) corresponds to the initial condition (t = 0) –the distribution function is a delta function at the origin.The next shown stage corresponds tothree steps inN or time. The chain has three monomer units or the particle hasdiffused bythree “steps”. The next shown stage corresponds to twelve steps - the chain is even longer orthe particle has diffused for a longer time.

3 Phenomena of Polymers

While we have focused on the conformation of a single chain inthe last chapter we now willwork out phenomena of polymers. There we will describe effects of polymers in dilute solution– the chains are isolated – in semidilute solutions and in blends. Whereas the first state obviouslycan be treated as idealized, the last state behaves also quite ideal. The semidilute solution is themost complicated state and will be treated in the last section.

3.1 Polymer near a hard wall

The modelling of a polymer near a hard wall bases on the Green function of section 2.8 [18].We assume the polymer to be in dilute solution, which means that the polymer coils do notintersect. Furthermore we assume an idealization which is true for θ-solvents. The detailed


Fig. 12: (left) The polymer in the presence of a hard wall potential. (right) Disregarded con-formation corrected by an allowed conformation. This correction is done with the method ofimages.

meaning is discussed in section 3.4. For the present modelling it simply means that we can usethe Green function as already derived.The Green function of a totally free polymer is already givenin equation 56. Now the presenceof a plain wall limits the polymer to the half spacez > 0 only. A possible attractive or repulsivepotential in the vicinity of the wall is not considered here –the boundary condition just prohibitsthe presence of the polymer beyond the wall. This also implies thatG = 0 at the boundary forz = 0 or z′ = 0. One could imagine to solve the Green function for a potential being large orinfinity in negative half spacez < 0 (see Fig. 12, left). But fortunately a method developedfor electrodynamics yields a simple solution. By the methodof images (image charges) onedirectly obtains a solution with the correct boundary condition:

G(~R, ~R′, NK) = (23πNKℓ2

K)−3/2 · exp

(

− 3

2NKℓ2K

(x − x′)2

)

exp

(

− 3

2NKℓ2K

(y − y′)2

)

(65)

·[

exp

(

− 3

2NKℓ2K

(z − z′)2

)

− exp

(

− 3

2NKℓ2K

(z + z′)2

)]

While this solution is formally correct a plausible interpretation should be given (Fig. 12, right)furthermore. The solution with the initial vector being in the correct half space (z > 0) and thefinal vector being in the wrong half space is prohibited. The mirrored solution of the final vectoradds one possible path where the polymer touches the wall andmakes the originally consideredpath impossible. Possible paths contribute positively while impossible paths cancel out. To getan idea of the polymer concentration near the wall, one has tointegrate the Green function overpossible end-vectors~R′ for a given initial vector.

ρ(z) =

∫

d3R′ G(~R, ~R′, NK) (66)

= (23πNKℓ2

K)−1/2

∞∫

0

dz′[

exp

(

− 3

2NKℓ2K

(z − z′)2

)

− exp

(

− 3

2NKℓ2K

(z + z′)2

)]

= erf

(√

3

2· z

Ree

)

The two Gaussian integrals inx andy-direction directly cancel out with the normalization. Theintegral in positive half space yields the error function. This solution is depicted in Figure 13.


0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

z = Rg

z / Ree

ρ (z)

z = Ree

Fig. 13: The distribution of an end-monomer relatively to a hard wall. The bulk concentrationis normalized to be unity. The two distancesRee and Rg correspond to≈ 10% and≈ 50%deviations of the bulk value.

While for a distance of the average end-to-end vector the concentration of the polymer deviatesby≈10% from the bulk value only, the distance of the average radius of gyration approximatelypredicts where the concentration reaches the half-maximumvalue. The normalization of thebulk value to unity is due to the missing normalization of theinitial vector (see eqs. 58 and 63).

It is more likely to find the polymer in the bulk phase than close to the wall. The polymerobviously is depleted by the wall. Near a wall, the polymer has less allowed configurations.This means a loss of entropy close to the wall, which is unfavorable. Therefore, the depletioneffect is of entropic origin.This depletion effect is important for a mixture of dissolved colloidal particles and polymers.The effective depletion zones of isolated colloidal particles is large, while for closely packedcolloidal particles the depletion zones overlap and therefore are effectively smaller. Thereforethe polymer depletion can lead to a phase separation of colloidal particles and polymers [19].This effect is highly interesting for the formulation of paints for instance.

3.2 Rubber elasticity

Rubber consists of high molecular weight polymers [5, 20]. To obtain a stable rubber themolecules are chemically cross-linked in a vulcanization process (see Fig. 14) and a polymernetwork is obtained. Rubber can be strained by several 100%,and its original shape is reversiblyobtained when the stress is released. The elastic modulus increases linearly with temperature.Only at very low temperatures the elastic modulus becomes comparable to normal solids andthe rubber becomes fragile. The deformation is defined by theelongation say inz-direction (seeFig. 15).


Fig. 14: Principle of a polymer network. (a) non-vulcanized – just topological cross-links arepresent, (b) vulcanized – some cross-links are chemically bond.

Fig. 15: The polymer network is stretched in one dimension by a factorof λ. Due to theconservation of volume the polymer contracts in the other directions.

λ ≡ λz = L/L0 (67)

It is a very good approximation, that the volume of a rubber isconserved under strain. There-fore:

λx · λy · λz = 1 (68)

Usually the rubber is isotropically deformed in the perpendicular directions and contracted dueto the conservation of volume.

λx = λy = λ−1/2 (69)

Now we assume that the positions of all cross-links are uniformly transformed by the sametransformation. This behavior is called affine. The transformed coordinates read then:

(x′, y′, z′) = (λ−1/2x, λ−1/2y, λz) (70)

As the thermodynamic potential we consider the free energy difference. The difference consid-ers the stretched relatively to the non-stretched rubber.


∆F (λ) = F (λ) − F (λ = 1) = ∆U(λ) − T∆S(λ) (71)

It is obvious that the free energy has energetic and entropiccontributions. At low temperaturesthe energy becomes dominant, which is mainly determined by the monomer properties. Thisenergy is responsible for the behavior similar to normal solids. At high temperatures the entropyis dominant, and we will focus on this contribution. The Green function (see eq. 56) of a chainsegment between two cross-links is then given by:

P (~R′) = (23πR2

ee)−3/2 exp

(

− 3

2R2ee

(λ−1x2 + λ−1y2 + λ2z2)

)

(72)

The used coordinates are the stretched coordinates (eq. 70). From this Green function (orprobability) the free energy can be calculated.

∆F (λ) = −T∆S(λ) = −kBT · n 〈ln P (λ) − ln P (λ = 1)〉 (73)

= 12kBT · n · (2/λ + λ2 − 3)

The entropy is dominant. To account for the number of chain segments, the numbern is multi-plied with the single segment contributions. The averagingover the squared coordinates cancelswith the prefactor of the argument in eq. 72. The force due to the strain is calculated by thederivative of the free energy:

∂∆F/∂L = kBT · n · (λ − λ−2)/L0 (74)

The stress is obtained by the force normalized to the area, and one obtains:

σz(λ) = kBT · n · (λ − λ−2)/V0 = RT · ρ/MC · (λ − λ−2) (75)

The final result is slightly rewritten by expressing the number of chain segments asn =NAρV0/MC . The gas constant isR = kBNA with NA being the Avogadro number. The factortemperature is responsible for the higher elastic modulus at elevated temperatures. The densityρ being always of the order of 1g/cm3 does not change the elastic modulus. The molar mass of achain segment between two cross linksMC is an important factor. Highly cross-linked rubbers,i.e. MC small, are harder.The idealized and real dependence of the stressσ on the strainλ is discussed in the context ofFigure 16. For stresses around 1 and below the theory fits the experimental data nicely. Forintermediate strains the stress is lower than theory. This is due to topological cross-links whichcan slide away upon external stress. Another factor is the fluctuations of the cross-link positions,which we assumed to be transformed strictly. These fluctuations also account for a decreasedstress. At even higher strains the chains do not behave Gaussian anymore – the chain segmentsare strongly stretched – and therefore a higher stress is found than in our simple theory.The origin of rubber elasticity is entropy, and the elastic modulus increases with temperature.In this context we can go back to the Hamiltonian of eq. 59. Theelastic constantk(T ) alsolinearly increases with temperature, and can be seen as the origin for the rubber elasticity. Theintermediate filaments of the cytoskeleton form also a network in the cell, which is responsiblefor the reversible deformability of a cell. It can be discussed if this elasticity is purely entropic,but surely the cross-link density (orMC) would affect the rigidity of the network.


Fig. 16: Stress strain curve of a rubber [21]. The details of this curve are discussed in the text.

3.3 Scattering of chains

The scattering techniques make up the best way to observe molecular (and atomic) structures.2

One uses a certain kind of radiation (light, x-rays, neutrons), which usually includes manywavelengths. A monochromator selects a certain wavelengthλ. The divergence of the beamis then usually reduced by apertures (collimation). This well defined radiation hits the sample.While usually larger amounts of the radiation are transmitted without being diffracted, the in-teresting information about the sample structure is obtained by the scattered intensity. Largearea detectors allow to collect as much information as possible. The intensity is measured asa function of the scattering angleθ, which can be expressed in terms of a momentum transferQ = 4π sin(θ/2)/λ. The momentum transferQ is the variable of the Fourier space, which wealready introduced in section 2.6. The scattering intensity is proportional to the form factorof a single chainS(Q) [6, 7, 22]. The form factor of a single chain considers interferencesof all possible pairs of “monomers”. For the time being the “monomer” units shall build up aGaussian chain. Further refinements are discussed below.

S(Q) ∝ 1

N

N∑

i,j=0

⟨

exp(

i ~Q · (~Ri − ~Rj))⟩

(76)

∝ 1

N

N∑

i,j=0

exp

⟨

−12

(

~Q · (~Ri − ~Rj))2⟩

(77)

2Further details of a scattering experiment are discussed inthe contribution of W. Schweika B1. We restrictourselves to the scattering of polymer chains.


∝ 1

N

N∑

i,j=0

exp⟨

−16~Q2 · (~Ri − ~Rj)

2⟩

(78)

Equation 77 has moved the averaging from outside to the inside of the exponential. This Gaus-sian transformation in principle uses similar arguments aswe used between equations 50 and55, and it becomes exact for Gaussian chains. The next rearrangement just makes use of theaverage projection (see also eq. 53). In the present stage weuse the property of a Gaussianchain and replace the summation by an integration, which is correct for large enough lengthscales (see discussion below).

S(Q) ∝ 1

N

N∫

0

di

N∫

0

dj exp(

−16~Q2 · |i − j| · ℓ2

K

)

(79)

= N · fD(Q2R2g) (80)

fD(x) = 2x2 (exp(−x) − 1 + x) (81)

In principle the same arguments were used for eq. 33. The finalresult yields the Debye func-tion fD, which is often observed in scattering experiments on polymeric systems. It has twoimportant regions, namely for small and relatively large scattering vectorsQ.

S(Q) ∝ N(1 − 13Q2R2

g) for smallQ (82)

∝ N · 2/(Q2R2g) for largeQ (83)

For small scattering vectors one obtains a maximum intensity at Q → 0, which contains thenumber of correlated monomersN . Experienced researchers prefer to use the molar volume thata polymer chain coversS(Q → 0) ∝ V . The dependence of the intensity at small scatteringvectors reflects the radius of gyration. So at small scattering angles one obtains informationabout the chain as a whole molecule. At larger scattering angles, one obtains aQ−2 dependencewith a prefactor, which depends on a segment lengthℓσ. This segment length is defined by theratio of the radius of gyration and the degree of polymerization 6R2

g/N = ℓ2σ, and compares the

real chain with the idealized freely jointed chain. IfN is defined with respect to a backbonebond as the basic unit, one obtainsℓσ =

√C∞ ℓ. For details further reading in reference [2]

is recommended. Therefore, at larger scattering angles oneobserves intra-chain parts. Theself-similarity of chain parts on different length scales is observed. Also information about therigidity C∞ is obtained.The described behavior can be fully observed from a small angle neutron scattering experimenton polymers [23]. An example for polystyrene is shown in figure 17. At small scattering vectorsQ < 3 · 10−3A−1 the intensity levels off. Since the molar volumeVW is relatively large, the fulloff-levelling is not observed in this experiment. At largerscattering vectors one nicely observestheQ−2-behavior, which tells about the self-similarity of the chain over many length scales.One interesting detail of this experiment is that the sampleconsists of protonated and deuteratedpolymer only. When discussing the scattering of a polymer blend then interferences betweendifferent molecules occur as well. These contributions have the sameQ-dependence as a singlepolymer [7, 24], and so the scattering of a polymer blend looks like the scattering of isolatedpolymers (in aθ-solvent, see section 3.5).


Fig. 17: Scattering of a polystyrene / deuterated polystyrene blend[23]: The structure factoras a function of the scattering vector.

The whole calculation assumed “small” scattering vectors,which are small compared to the in-verse monomeric sizeQ ≪ ℓ−1. Approaching this limit would allow one to observe a real rigidpart of the chain. At even larger scattering vectors one would observe the chemical structure, i.e.each atom within the monomer. The wholeQ-range would therefore span the following sizes:whole chain – self-similarity of chain – rigid part of the chain – atomic structure. There existapproaches, which cover the rigid part of a chain as well [25,26] (there applied for polystyrene).More interesting are these theories in studies of worm-likemicelles [27], where the apparentQ-range is shifted to lower values.The for all scattering experiments interestingQ-ranges can be also discussed in the contextof the continuous description (eq. 63) (see contribution R.Winkler C3 and [28]). The firststage of approximation, which is presented in this lecture,just considers the Gaussian chainwith ∂ ~R/∂s terms. The only parameter of this chain model is the end-to-end distance〈R2

ee〉.The correspondingQ-range covers the whole chain and the self-similarity of thechain. Forthis Q-range the expression of the end-to-end distance can base onany segment lengthℓ. Thenext stage of approximation considers a semiflexible chain with additional∂2 ~R/∂s2 terms inthe hamiltonian. The correspondingQ-range additionally covers the rigid part of the chain.The contour lengthL comes into play, which makes a segment length distinguishable. Thenatural segment length of such a chain is given byℓ = 〈R2

ee〉/L, and the natural degree ofpolymerization is given byN = L2/〈R2

ee〉. A higher stage of the continuous description wouldintroduce the torsional energy. For the present lecture we can safely neglect the higher stages.

3.4 Deviations – The Flory Exponent

Differences from the behavior above occur when polymers aredissolved in good solvents [8].An example of scattering curves is shown in Figure 18. For this example the behavior at large


Fig. 18: Scattering of a polystyrene solution in deuterated benzene[23]: The structure factoras a function of the scattering vector.

scattering vectors is more likeQ−1.5. This behavior extends over a largeQ-range and occursfor many samples, which speaks for the significance of the effect. In viscosity measurementsone observes the end-to-end distanceRee as a function of the degree of polymerizationN withdeviations from the ideal coil behavior (eq. 41). Generallyone can write for the end-to-enddistance:

⟨

~R2ee

⟩

= N2ν · ℓ2e ν ∈ [1

2..1] (84)

The exponentν lies in the range between12

and1, which corresponds to ideal coils and rods.The exact value of polymers in good solvents is the issue of this section. For curiosity therelation to the radius of gyration should be mentioned as well:

⟨

~R2g

⟩

=ℵ6

⟨

~R2ee

⟩

with ℵ =6

(2ν + 1)(2ν + 2)(85)

The prefactorℵ is of the order of unity, and was calculated theoretically inreference [3]. Itagrees with our results in the limits of ideal coils and rods (eqs. 38 and 40):While we considered the local rigidity of a chain – and this can be a result of steric hindranceof side groups – we neglected hindrance of chain parts with very different indicesn andm(see Fig. 19). This interaction is called the excluded volume interaction. The Flory argumentallows to calculate the exponentν within a very good approximation. While we already lost theprecise reference of some numbers (sayN) to the polymer unit (be it monomers or backbonebonds) sometimes, we now will deliberately argue on the basic dependence. This arguing iscalled scaling argument then.The occupied volume of a diluted, isolated polymer chain ind dimensions is thenV ∼= Rd. TheradiusR is a typical size of the chain. The monomer density in this isolated volume is given byρ ∼= N/Rd then. Now the average repulsive energy can be calculated by:


Fig. 19: Whereas the chain flexibility results from steric hindranceof neighbored monomers,the excluded volume interaction bases on repulsive interactions of distant monomersn andm.

Table 2: Comparison of the Flory exponent based on the Flory argumentwith exponents fromrenormalization calculations. The agreements are surprisingly good.

d νF ν (best value)

1 1 1 exact2 3/4 3/4 exact3 3/5 0.5884 1/2 1/2 exact

Erep∼= ε · V · ρ2 ∼= ε · N2/Rd (86)

The expression corresponds to a Coulomb interaction of a (homogenous) charge densityρwithin a volumeV . The factorε just takes care for the correct unit of the repulsive energy.Now the entropy of a chain can be calculated in the manner of previous calculations (see eq.73). It basically is the logarithm of the Green function:

S ∼= kB · ln P (R) ∼= −32kBR2/(Nℓ2) (87)

The free energy is given by the energy and entropy then:

F = Erep − TS ∼= ε · N2/Rd − 32kBTR2/(Nℓ2) (88)

The minimization of the free energy is done via the first derivative∂F/∂R = 0. The expressioncan be sorted byR andN-terms then, and one finally obtains:

R ∝ Nν with ν = 3/(2 + d) (89)

This result can be compared with more sophisticated calculations with the formalism of a renor-malization group (see Table 2). It is very astonishing that the simple scaling argument of Florydelivers the exact values except for 3 dimensions. The more precise value ofν = 0.588 onlyslightly deviates from the Flory value0.6. The deviation from the ideal coil behavior (ν = 0.5)means, that the polymer swells due to the repulsive excludedvolume interaction.


Fig. 20: A Flory chain with sections of the diameter∅. The scattering power law is derived inthe text.

The connection to scattering experiments is drawn in the following (see also Fig. 20). A chaincan be divided into 3-dimensional sections of diameter∅. Within such a section a sub-chainis found with the Flory-behavior (eq. 84), and so the number of monomers of the sub-chain isNν

∅∝ ∅. The scattering intensity is proportional to the numbers ofmonomers within such a

sub-chainS(Q) ∝ N∅, and the respective scattering vector is connected with thediameter viaQ ∝ ∅

−1. Finally, one obtains for the scattering at large enoughQ-values:

S(Q) ∝ Q−1/ν (90)

Therefore the Flory exponent can be observed in a single scattering experiment. This is amuch simpler way than measuring the end-to-end distance by viscosity for several molar masspolymers (see eq. 84).It shall be mentioned that the Flory exponent occurs for polymers in good solvents. There,the contact of the polymer with itself is unfavorable (Fig. 19), which leads to the effect de-scribed above. But solvents exist (θ-solvents), for which the interaction solvent-polymer is inequilibrium with the interaction polymer-polymer. Energetically, there is no difference for thepolymer-polymer interaction then, and the ideal coil behavior is retrieved (ν = 1

2). In a polymer

blend there is no unfavorable interaction neither, and one obtains the ideal coil behavior as well,as we will also see in the next section.

3.5 Semidilute Solutions – The Blob Concept

The characteristic issue of semidilute polymer solutions is the contact between different poly-mers, which leads to many topological cross-links. Therefore, such a solution behaves gel-like.The cytoskeleton certainly is a semidilute polymer solution with topological cross-links. Par-tially, aspects of biological cross-linking play an additional role. Furthermore, semidilute solu-tions are interesting for glues. A poor mans glue consists only of polymer and solvent. Whenthe solvent evaporates the polymer hardens and fixes two pieces together. More sophisticatedglues consist of a ready made polymer and a solvent, which is amonomer. This monomer canbe polymerised, and after full conversion the whole glue is hardened.We will derive expressions for the blob concept of DeGennes [9, 29] in the following, againin the manner of scaling arguments. The semidilute polymer solution is a network, with manypolymer-polymer contacts of a typical distanceξ (see Fig. 21). Within such a blob the polymer


Fig. 21: A polymer network formed by a semidilute polymer solution. Ablob is defined by theintersection of a chain segment with other chains. The numbers ξ andg are related in the text.

behaves like in a good solvent, and so one findsg monomers inside the blob according toξ ∝ gν

with ν = 0.6. Beyond the blob the correlation is lost (ξ is also called correlation length), andone says that the excluded volume interaction has been screened out. Beyond the blob thepolymer behaves like an ideal coil:

Rg =√

3 · ξ ·(

N

g

)1/2

(91)

The fractionN/g is the number of blobs, which behave like an ideal coil, andξ plays a role ofa segment size. For low concentrationsφ < φ∗ the blob becomes large, and finally covers a fullsingle polymerg ∼= N . Then, we recover the situation of a polymer in a good solvent(section3.4). For very high concentrationsφ ∼= 1 the blob sizes shrinksg ∼= 1 and a blob contains amonomer approximately. Then we recover the situation of a polymer blend (compare section3.3, Fig. 17). The overlap concentrationφ∗ is reached, when the monomer concentration of anisolated polymer reaches the overall polymer concentration:

φ∗ ∝ N · R−3g ∝ N1−3ν (92)

The respectiveN-dependence has been used then. Now the correlation length is expressed inthe manner of a scaling argument:

ξ ∝ Rg

(φ

φ∗

)y

(93)

At the overlap concentrationξ has to agree with the radius of gyration, and at higher concentra-tions it behaves with the concentration likeφy with an unknown exponenty. Another scalingargument says, that the blob-size does not depend on the degree of polymerization if the con-centration remains constant. The polymer is assumed to be much larger than a blob and thenthere is no influence on the blobs anymore. From theN-dependence one obtains:


ξ ∝ Nν

(φ

N1−3ν

)y

∝ N0 (94)

And finally one can solve for the unknown exponenty:

ξ ∝ φy y = ν/(1 − 3ν) = −3/4 (95)

This result can be used to derive the same dependence for the radius of gyration. One simplygoes back to equation 91 and uses all known dependencies:

R2g ∝ ξ2N

g∝ ξ2−1/ν (96)

∝ φ(1−2ν)/(3ν−1) = φ−1/4 (97)

The two dependencies ofξ andRg as a function of the concentrationφ were observed experi-mentally by neutron scattering [29] (see Figs. 22 and 23). For the measurement of the correla-tion length one uses protonated polymer in deuterated solvent. Since correlations are screenedout beyond the blob, one just observes the blob-scattering,the scattering from the inside of theblob. The measurement of the radius of gyration is more tricky. One uses small amounts ofdeuterated polymer amongst protonated polymer dissolved in a normal solvent (equivalent toprotonated polymer). Then the neutrons are sensitive to thesmall amounts of deuterated poly-mer, whose chains appear to be isolated. At small scatteringvectors one observes whole chainproperties such as the radius of gyration. The presented experiments agree well with the blobmodel of DeGennes.

4 Conclusion

In chapter 2 we have derived the chain conformation in terms of observables like the end-to-enddistance and the radius of gyration. We developed several theories, some of which includedrigidity. All models agree in the general conclusion that the typical sizeR scales with

√N ℓ

for ideal coils. N is the degree of polymerization andℓ is a typical size for the monomer (orbackbone length). On the other hand the typical size of rigidrod molecules scales likeNℓ,which is the contour length. The model of semiflexible chainsallows to interpolate between thetwo limits.The distribution of the end-to-end distance is a Gaussian, independent of the distribution ofa single segment. This is due to the central limit theorem. The distribution of the end-to-end distance is a Gaussian function as shown for the discreteand continuous description. TheGreen function of polymers has similarities with the diffusion process, and quantum mechanicsin imaginary time.In chapter 3 we described phenomena of chain molecules. The underlying theories alwaysbase on the developed concepts of chapter 2. The depletion effect in colloidal systems withdissolved polymer plays an important role for formulationsof paints. The elasticity of polymernetworks is important for intermediate filaments of the cytoskeleton, and for rubber. Scatteringis of special interest for the observation of polymers, since neutrons and x-rays are sensitiveto structural sizes of polymers. Deviations of polymer conformations are observed in goodsolvents – the excluded volume interaction leads to polymerswelling. Polymers in semidilute


Fig. 22: Neutron scattering experiment on semidilute polymer solutions [29]. The parameterξ(single blob) is accessible by labelling the polymer network against the solvent.

Fig. 23: Neutron scattering experiment on semidilute polymer solutions [29]. The parameterRg of the whole chain is accessible by labelling a few polymers against the polymer networkand the solvent.


solution form networks which are interesting for the cytoskeleton and glues. The here developedpictures and concepts are widely applied for many other phenomena of polymers, and thereforemay give the reader a solid fundament in the field of polymer physics.

Appendices

A Appendix: Theorem of Lagrange

The theorem of Lagrange is used to eliminate the coordinates~si relatively to the center of massin the expression for the radius of gyration. Each such vector can be reformulated by the initialvector~s0 plus a chain segment vector~r0,i:

~si = ~s0 + ~r0,i (98)

The square within the radius of gyration expression is then expanded:

~R2g =

1

N + 1

N∑

i=0

(~s0 + ~r0,i) · (~s0 + ~r0,i) (99)

= ~s 20 + 2 · (N + 1)−1~s0

N∑

i=1

~r0,i + (N + 1)−1

N∑

i=1

~r 20,i (100)

Now one should insert, that the sum of all center of mass coordinates is zero. This can beexpressed in other ways using eq. 98:

N∑

i=0

~si = 0 ⇒ ~s0 = −(N + 1)−1

N∑

i=1

~r0,i (101)

~s 20 = (N + 1)−2

N∑

i,j=1

~r0,i · ~r0,j (102)

For the expanded expression of the radius of gyration one obtains then:

~R2g = −(N + 1)−2

N∑

i,j=1

~r0,i · ~r0,j + (N + 1)−1

N∑

i=1

~r 20,i (103)

Another insertion is the law of cosines:

~r0,i · ~r0,j = 12(~r 2

0,i + ~r 20,j − ~r 2

i,j) (104)

This finally allows to reformulate the radius of gyration further. After completing the sums torun from 0 instead of 1 again, one finally obtains the desired result:


~R2g =

1

2(N + 1)−2

(N∑

i,j=1

~r 2i,j + 2

N∑

i=1

~r 20,i

)

(105)

=1

2(N + 1)−2

N∑

i,j=0

~r 2i,j (106)

= (N + 1)−2∑

0≤i<j≤N

~r 2i,j (107)

This theorem is not to be confused with the theorem of Steiner/Huygens, which only considersthe expansion (eq. 100) with the center of mass coordinates (eq. 101).


References

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[2] R. G. Larson,The Structure and Rheology of Complex Fluids(Oxford University Press,Oxford, 1999)

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[29] M. Daoud, J. P. Cotton, B. Farnoux, G. Jannink, G. Sarma,H. Benoıt, R. Duplessix, C.Picot, P. G. DeGennes, Macromol.8, 804 (1975)

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