Figure 4.6: One dimensional conditional monomer densities %^
(x
=R
;S
) as functions of the
scaledvariablesu
=x
=R
,todemonstratethebreakdownofcondition(4.36) inastrict sense. The
curves weresampled forchainlengths N =1000.
0 0.1 0.2 0.3 0.4
0 0.5 1 1.5 2 2.5 3 3.5 ρ ~ 1 (x 1 /R 1 )
x 1 / R 1 a)
ρ ~
1 (x 1 /R 1 ) N=1000 N=300 N=100 N=30
0 0.1 0.2 0.3 0.4
0 0.5 1 1.5 2 2.5 3 3.5 ρ ~ 2 (x 2 /R 2 )
x 2 / R 2 b)
ρ ~
2 (x 2 /R 2 ) N=1000 N=300 N=100 N=30
0 0.1 0.2 0.3 0.4
0 0.5 1 1.5 2 2.5 3 3.5 ρ ~ 3 (x 3 /R 3 )
x 3 / R 3 c)
ρ ~
3 (x 3 /R 3 ) N=1000 N=300 N=100 N=30
Figure 4.7: Eective scalingfunctions%~
asdenedineq. (4.41)fordierentvaluesofN. Thesolid
linesdenote theapproximateformsgiven ineq. (3.17a-c).
Althoughtheself{similarityassumptionisnottruelyvalid,itcanbeusedasagoodworkingtool
for the following reasons. The monomer densities enter the calculation of F
inter
via integrals,
cf. eq. (3.5). Forthese integralsto be correctly calculated, itis importantthat the size of the
region, where %(x;S;N) is non{neglible,scales properly. From g. 4.6 one can see that this is
indeedthecase. Therefore, forourpurposesitissuÆcienttoapproximatethevariousfunctions
shown ing. 4.6a-c for dierent S
by their average
~
We marked these averaged monomer densities by a tilde, since one can view them as
\ef-fective scaling functions" representing % (u)~ from eq. (4.36) (after integrating out the two
u
{coordinates orthogonal to u
). Note that we suppressed the N{argument in the
deni-tion (4.41), since,as already mentioned, the dependence onN predicted by eq. (4.36)is valid.
This validity can also be inferred directly from g. 4.7, where the data for N = 30;100;300,
and 1000 can not be distinguished.
Thescaledmonomer densities%~
shown ing. 4.7canbeconveniently
approximatedbya superpositionof Gaussians, given ineqs. (3.17a-c). Theparameters
were determined by least squares ts and are given in tab. 3.2. While %~
1
) are quite well approximated by eqs. (3.17a) and (3.17c), respectively, eq. (3.17b) does
not providesuch agooddescription. Abetterexpression for%~
2 (u
2
) canbeachieved by adding
shifted Gaussian functionsas ineq. (3.17a),and isgiven explicitlyin appendix B.3. However,
we decided to deal with eq. (3.17b), because the better description increases the computation
time of F
inter
by a factor of four, but does not much improve the overall accuracy of %(x;S).
We note that the functions %~
(u
) in eqs. (3.17a-c)should not be confused with the function
(x;N) (orits \components" after integrating out two coordinates) that has been considered
earlier by Janszen etal. [Jan96].
Inordertospecifythemultivariatescaleddensityweemploy,asforP(S)before,aseparation
ansatz. This yieldsfor %(x;S;N) from eq. (4.36)the nalresult
%(x ;S;N)=(N+1)
) are taken from eqs. (3.17a-c). The separation ansatz may be used also
withoutinvokingtheself{similarityassumption(4.36)inordertoreducethe complexityof the
multivariate density %(x;S;N).
Totest the validity of the separation ansatz we denefor each Gaussianchain by
v
the (scaled)mean moduliof the monomer coordinates inthe principal axissystem, and check
whether the correlation coeÆcients
0 0.5 1 1.5 2
0 0.5 1 1.5 2
x 2 / R 2
x 1 / R 1 ρ
~
12 (x 1 /R 1 ,x 2 /R 2 ) a)
0 0.5 1 1.5 2
0 0.5 1 1.5 2
x 3 / R 3
x 2 / R 2 ρ
~
23 (x 2 /R 2 ,x 3 /R 3 ) b)
0 0.5 1 1.5 2
0 0.5 1 1.5 2
x 1 / R 1
x 3 / R 3 ρ
~
31 (x 3 /R 3 ,x 1 /R 1 ) c)
Figure 4.8: Contour plots showing the comparison of %~
(x
=R
)%~
(x
=R
) (straight lines) with
numerical data of %~
(x
=R
;x
=R
) (dashed lines). The isolines are drawn at function values,
which areinteger multiples of0:02.
are much smaller than one. Note that, because of the symmetry %~
considered the momentsof the moduliin eq. (4.44). As shown in tab. 4.2, we nd j (v)
j to be
less than 5% inthe simulations. Also,when comparing the two{variatedensities
~
with the two{foldproducts %~
) ing. 4.8 a satisfactory agreement is obtained.
As an ultimate test of the separation ansatz we sampled the full three{variate scaled
monomer density %(u )~ in the simulations. To this end, we subdivided the u = x=R space
intosmallboxes of length u
=0:07 and determined the mean number of monomers ineach
box by averaging over Gaussian chains irrespective of their S . Then we calculated the
rela-tive error between the separation ansatz Q
) from eqs. (3.17a-c) and the
simulated % (u)~ as a function of % (u)~ (for all u). We found that the relative error is conned
to a narrow band of maximal 25% discrepancy except for very small %(u).~ These small % (u),~
however, are not importantinthe calculationof the overlap integralsin eq. (3.5).
The monomer density %(x ;S;N) specied in eq. (3.16) allows us to calculate the overlap
integralsandthustodetermine F
inter
fromeq.(3.5) analytically,asexplainedinappendix C.1.
4.2.3 Cylindrical Densities
Determining the monomer density with respect to the center of mass only, yields a spherical
shape. In the principalaxis system a bimodal, dumbbell like shape was found. Now, dening
one axisr
AB
between the twocentersof masscorresponding tothe two blocksof thechain and
regarding the monomer density with respect to this axis, wehave cylindricalsymmetry.
Sinceweareinterestedindetermining%
X
),describingoneblockofthechain,
we willinvestigatethe formof themonomer density ofone block paralleland perpendicularto
r
AB
. We willrestrict the discussion to the case of equallysized blocks (N
A
=N
B ).
At rst glance one might think that the monomer distribution of one part of the chain is
independent of the state of the other part, because all links are uncorrelated. Therefore it
should besphericallysymmetric withrespect toitsown center of mass. However, as the chain
parts are connected atthe ends, there is acorrelation between the position ofthe lastbead of
part A and the center of mass of the other part r
B
. Thus strictly speaking
%
) is dened in eq. (4.30). This can be seen in g. 4.9, where projections of
the scaled conditional monomer density %
A
) are shown. The origin is dened
Table 4.2: Testof theansatz eq. (3.16) with correlationcoeÆcientsdened ineq. (4.44).
(hv
0 0.2 0.4 0.6 0.8
-2 -1 0 1 2 3
ρ A,|| (r || ; r 2 AB ) / (N+1)
r || / R A (a)
0 0.5 1 1.5 2
0 0.5 1 1.5 2
ρ A, ⊥ (r ⊥ ; r 2 AB ) / (N+1)
r ⊥ / R A (b)
Figure 4.9: Projections of the conditional monomer density %
A (x;r
AB
) of one block (a) parallel
and (b)perpendicularto r
AB
. The respective Gaussianfunctions,eq. (3.7), aredraw forcomparison
(straight lines). The chain lengthsare N
A
=N
B
=50.
by the center of mass of block A, the parallel direction by r
AB
and the center of block B is
located in positive r
k
direction. The data has been scaled with respect to R
A
, cf. sec. 4.2.1,
and averaged with respect toR
B
. To obtain the dependency on r 2
AB
, shown by dierentlines,
a procedure analogous to the determination of the data for
r (x ;R
2
G
) in sec. 4.2.1 has been
carried out. However, here P
r (r
2
AB
) has to be replaced with P
R (R
2
G
) in eq. (4.32). Again the
dierentgraphs have very dierent statisticalweights.
WhilemostcongurationsaredescribedwellbytheGaussianapproximation,eq.(3.7),there
are deviations for very large values of r 2
AB
. On one hand with increasing r 2
AB
the maximum
of %
A;k (x;r
AB
) shifts tosmaller values of r
k
and a shoulder of the density emerges in opposite
direction, g. 4.9a. On the other hand,the extension ofthe density indirectionperpendicular
to r
AB
shrinks, g. 4.9b. Therefore the block density is elongated with respect to r
AB and
shortened inperpendiculardirection. However, the overall scaleisnot drasticallyaltered. The
extension is still of order R
A
, only the positions, where the densities go to zero, are changed
by a factor of at most two for the values observed. Therefore the Gaussian approximation,
eq. (3.7), isa reasonable startingpointfor the disphere model.
4.3 Origin of the Bimodal Form
Inthis section webriey addressthe originof the bimodalshapeof the monomerdensity with
respect to the largest principalaxis. As the bimodalshape occurs alsoin the average density
%, we willdiscuss this simplercase.
Janszenetal.haveanalyzedtheshapeofarandomwalkintheprincipalaxissystem[Jan96].
Further on, they separated the chain into two equal blocks, dening the rst block A and the
second block B by the positions of their centers of mass r
A and r
B
, and recorded the average
monomer density of block A in the principal axis system of the whole chain, nding an egg{
likeshape. They pointed out, that this shape is observed only in the principalaxis system of
the total chain, while in the principal axis system of block A the bimodalform would emerge
again. They claimed,that the naturalsegregation of the chain intotwoparts (inwhichcase a
denition of arst and secondblockis unambiguouslypossible)isresponsible forthe bimodal
shape. We willrene this statementin the following.
Therefore we consider a Gaussian chain not with respect to its principalaxis system, but
with respect to the axis r
AB
= r
B r
A
. There is a strong correlation between r
AB
and the
rst principalaxis. This can beseen directly by determiningthe distributionof theprojection
between the two normalizedvectorseither numericallyorfromthe followingsimpleargument:
When considering the chain as a disphere, consisting of two equal parts with average radii of
gyration hR 2
A
i=hR
G
i=2, we can calculatean average squared radius of gyration hS
k
i parallel
to r
AB
fromthe theorem of SteinerhS
k
i=3. Ananalogous result
for a perpendicular directionyields hS
?
i=4,the average
extension of the random walk in direction perpendicular to r
AB
is only half its extension in
paralleldirection, avery directargumentwhy typicallythe randomwalk isnot spherical. The
inertia moment of a rigid body with respect to an arbitrary axis in space, characterized
by its normalized projections on the principalaxis system (r
1
;r
2
;r
3
), is given by an equation
describing an ellipsoid,
=
The eigenvaluesof the inertiatensorare denoted by
. Dividingeq.(4.47)bythe mass Nm,
taking the average and neglecting correlations,we nd
hS
Knowing hS
k
i, cf. tab. B.1, we can estimate the typical squared
projection of r
AB
on the rst principal axis. Noting hr 2
i2[0:83;0:86]1=3,wherethelastvalueisthereferenceforrandom
orientation. The numeric value hr 2
1;k
i = 0:85 conrms the estimate. Therefore r
AB
is almost
parallel tothe rst principalaxis and we can discuss the bimodalshape with respect tor
AB .
1
Because the typical extension of a block is given by hR 2
A
i = N=12 and the typical distance
between the two centers of mass is hr 2
AB
i =N=3 the criterion for the occurrence of a bimodal
form
Bydividing the chain not intotwo but intofour equalparts and assigning the whole mass
of the four blocks to the respective centers of mass, not only one axis r
AB
, but a coordinate
system is dened. This is the principalaxis system of the foursphere model. Numerically we
nd, that the principal axis system of the foursphere model almost coincides with the exact
principal axissystem of the chain. Therefore a description of a Gaussianchain interms of an
ellipsoidalmodel and interms of a foursphere modelshould be verysimilar.
Finally, it is interesting, to compare gs. 4.9a and 4.6a, having the strong correlation
be-tweenthe two axesr
AB
and therst principalaxisinmind. The morepronouncedbimodality,
found in the principal axis system for large values S
1
, is reected in the monomer density
parallel to r
AB
for large r 2
AB
. The respective congurations on average consist of two balls at
each end with higher monomer density that are connected by a rod like region with smaller
monomer density, whereprincipally the bonds are oriented inone direction. The wholeshape
1
Numericalinvestigationsconrmthatthebimodalshapeisobservedwithrespecttor
AB .
is thereforedumbbell like and with increasing S
1 , or r
2
AB
respectively, the dumbbell stretches,
i.e.the rodelongates and the balls shrink.
This behavior is also reected in the results for %
r (x;R
2
G
) for large R
G
, cf. g. 4.5. In
this limit the value of R 2
G
is dominated by S
1
even more than on average, cf. g. 4.3, i.e.
R 2
G 'S
1
. The maximum of the monomer density shiftstohigher values of r=R
G
for large R
G .
Fromthe sphericallysymmetricpointofview, this behaviorisdiÆculttoexplain, becausethis
shaperesembles some kind ofhollowsphere. Butfromthe considerationsabove, this shiftcan
be interpreted by the elongation of the dumbbell, because the two balls of the dumbell are
responsiblefor the maximum in %
r (x;R
2
G
) inthe limitof high R
G .
4.4 Summary
Tosummarizethissection,wedeterminedtheinputquantitiesforthedierentmodels,dened
in chapter 3, and expressed them in approximate, but closed form. All approximations have
been tested extensively against numerical results. Despite approximate, the input quantities
capture the essential features found for Gaussian chains: The normal random walk scaling
with N,the strong asphericityof the chains inthe respective coordinatesystems, the bimodal
formofthe monomerdensity and itsrapiddecay beyond cutolengthsthat scalelinearly with
R
. Finally,the originof the bimodalform couldbe explainedby simpleconsiderations ofthe
disphere model.
To give a reasonable description of polymer systems, the models dened in chapter 3 should
exhibit several essential features. First of all, they should show the right scaling behavior
of various properties with chain length N in dense systems [Mur98]. Secondly, characteristic
features of polymer physics on the scale of R
G
, such as the correlation hole [Gen79] should
be found. Finally,the kinematic description for times larger
d
(but smallagainst the
hydro-dynamic regime)oughtto be correct.
In this chapter we will investigate properties of the Gaussian Ellipsoid Model (GEM) in
homogeneous and inhomogeneous bulk systems, and show that the GEM provides a suitable
description of polymer systems on the time and length scales intended.
5.1 Homogeneous Melt
To show that the GEM reproduces the most important semi{macroscopic properties of dilute
and dense polymeric systems, we investigate the following quantities: The scaling properties
ofthe radiusof gyration
R
G
uponN and c, thescalingbehaviorof thedistributionP
R (R
2
G
;N)
in a dense homogenous system of interacting ellipsoids, and the existence and scaling of the
correlation hole. Concerning the kinetic properties, we show that the ellipsoids show normal
diusion for times t >
d
and that other occurring correlation times are of order or smaller
than
d .
Allsimulationresultsinthis sectionarefor systemswithatleast M =1000ellipsoids. The
considered chain lengths are in the range N = 30;:::;400, therefore the systems correspond
to polymer systems consisting of order 10 5
monomers. Equilibrium averages were taken over
several thousand MonteCarlo Steps (MCS).
5.1.1 Static Properties
Asdiscussedinsec.3.2,the freeenergyasgivenineqs.(3.1{3.3,3.5)isaFlory{likedescription
of polymers. Accordingly, the gyrationradius of afree ellipsoid,that onlyinteractswith itself
but not with other ellipsoids,scales as R 2
G
N
6=5
[Gen79], see g. 5.1a.
In order to obtain the dependence on also, we redo some Flory type calculation, where
forsimplicityweconsider softspheres insteadofsoft ellipsoids. Therefore,the intra{molecular
freeenergy isbased onP
R (R
2
G
)ineq. (3.6)instead ofP(S),and the monomer density is given
by eq.(3.7) and the correspondingself{interaction termby eq. (3.8)instead ofthe expressions
for the GEM, but this willonly change prefactorsand not the scaling behavior itself. Putting
everything together, we obtain for the free energy of asingle self{interacting soft sphere
10 -1
Figure 5.1: a) Mean squared gyration radius of free (c ! 0), but self{interacting ellipsoids as a
functionof N,displayedforvariousvaluesoftheinteractionstrength . TheFloryresult
is shown for comparison. The inset shows the dependence on . The line indicates a power law
b()/ 2=5
. b) Scalingplot showingthecrossoverfrom Gaussto Florytype behavior.
F
The most probable value R 2
G;0
resultsfrom minimizingwith respect toR 2
This equation has the asymptoticsolution
10
Figure 5.2: Squared gyration radius
R 2
G
as a function of N in a dense (c = 0:85) system of M =
1000!4000ellipsoidsforvariousvaluesof(double{logarithmicrepresentation). Inallcasesalinear
behaviorisobserved.
R
in reasonable agreement with the numerical results for
free, self{interacting ellipsoids,see g. 5.1a.
Forsmall, oneshouldrecoverthe Gaussianbehavior,i.e.weexpect
(N). Thus, from continuity, we
nd
?
(N)N 1=2
. The corresponding scaling form
Wenowconsidersystemscomposedofmanyinteractingellipsoids. Asdiscussedinsec.1.1.3,
the Flory type scaling(5.3) changes to anormal randomwalk scaling if the monomer
concen-trationcexceedsthe overlapconcentrationc
?
. Thatthisis indeedthecase isshown ing.5.2.
Approaching c
?
frombelowwe can write (c
?
. Therefore we expect moregenerally the scaling form[Gen79]
for u1, todescribe the crossover from dilute
todensesystems. Accordingly,wehave
validityofthe overallscalingpredicted byeq.(5.4), weshowing. 5.3asanexample
as a function of cN 4=5
for a xed value =1:0. As can be seen from the gure, the data for
dierent cand N all collapse onto a commonmaster curve. We note that the scaling (5.4) is
followed only for N 1=2
. For N 1=2
by contrast, the ellipsoids would always exhibit
Gauss typebehavior.
5.6 10 -2 1.0 10 -1 1.8 10 -1 3.2 10 -1
0.1 1 10 100
— R 2 G / N 6/5
c N 4/5
N = 4000 N = 2000 N = 1000 N = 400
5.6 10 -2 1.0 10 -1 1.8 10 -1 3.2 10 -1
0.1 1 10 100
— R 2 G / N 6/5
c N 4/5
~ (c N 4/5 ) -1/4
Figure 5.3: Test of thescaling relation(5.4) for xed =1:0 (double{logarithmicrepresentation).
For small concentrations, thechains arealmost free and
R 2
G
saturates, corresponding to Flory{type
behavior. Intheregionof highconcentration theexpected scaling relationisalmost fullled.
0 1 2 3 4 5
0 0.2 0.4 0.6 0.8 1
P R (R 2 G ) N
R 2 G / N
ln(P R (R 2 G ) N)
R 2 G / N
Figure 5.4: ScalingbehaviorofthedistributionfunctionP
R (R
2
G
)inameltsystemforxedc=0:85,
=1:0, and variousN =30 (), 50 (), 100 (), 200 (N), and 400 (H). The solid linerefers to the
functiondenedineq. (4.10) withparameters a
R
=0:0665and d
R
=3:52. The insetshowsthe same
data insemi{logarithmicrepresentation.
-2 -1.5 -1 -0.5 0
0 0.5 1 1.5 2
( ρ ´ dist (r) - c ) N 1/2
r / N 1/2 N = 30 N = 50 N = 100 N = 200 N = 400
Figure 5.5: Scalingplot ofthe correlationhole forxedc=0:85 and =1:0 andvariousN.
Wefurthershowing. 5.4thatthe distributionfunctionP
R (R
2
G
;N)fordensesystems (cc
? )
obeys the same typeof scaling asfor non{interacting Gaussian chains, eq. (4.9), P
R (R
2
G
;N)
N 1
~
P
R (R
2
G
=N). In particular, the ansatz (4.10) can be used equally well for
~
P
R
(u) up to a
changeoftheparametersa
R
,andd
R
(seetab.3.1andthevaluesgiveninthecaptionofg.5.4).
It is worth noting that the correct scaling (5.4) in this case does not arise from the blob
picture, as described in sec. 1.1.3, because the blob size
m
is smaller than S
3
and does not
occurinthe ellipsoidmodel. Ratherthebehaviorarises fromcontinuitybetweenbothregimes.
To complete our study of the static properties of dense homogeneous systems, we discuss
the properties of the so{called correlation hole [Gen79]. Therefore we determine the mean
monomer density of all ellipsoids except the ith one as a function of the distance from the
center of mass of ellipsoid i,
% 0
dist
(r;N)= D
X
j6=i
% 0
j (y r
i
;N) E
: (5.5)
After performing the average in eq. (5.5), this mean monomer density of \distinctellipsoids"
is equal for all i and depends on r = jyj only. For large r&
R
G , %
0
dist
(r)!c, while for small
r
R
G , %
0
dist
(r) must be smaller than c due to the fact that ellipsoid i has been excluded
from the sum in eq. (5.5). Since 4 R
1
0 drr
2
[%
0
dist
(r;N) c] = (N+1), we expect a scaling
[%
0
dist
(r;N) c]N 1=2
f(rN 1=2
) for the correlation hole, in reasonable agreement with the
simulated resultsshown ing. 5.5.
5.1.2 Kinetic Properties
Nowwe discussthe dynamicalbehaviorof the system. As can be seen fromg. 5.6, the time{
dependent mean square displacement h[r
i
(t) r
i (0)]
2
i of anellipsoid exhibitsnormal diusive
behavior fortimes t>
d
. The disentanglementtime
d
can beidentiedwith the average time
an ellipsoidneeds todiuse overa distance
R
G
. The shorttime regime t<
d
is not of interest
here, sincewe donot intend tocapture thecomplicateddynamicsof polymersystems onthese
time scales, cf. sec. 1.1.4. In particular, the diusion coeÆcient
D= lim
t!1 h[r
i
(t) r
i (0)]
2
i
6t
(5.6)
willusually not exhibit the desired scalingwith N, e.g. DN 1
forRouse chains orDN 2
for entangled chains.
In fact, because D depends on r
max
, and r
max
is chosen to be some fraction K
r of
R
G N
1=2
, see sec. 3.4, the variation of D with N is inuenced by the simulation procedure
itself, i.e. by the value of K
r
. For the choice K
r
= 0:25 used in our simulations, D increases
with N, see g. 5.6, but other choices lead to a dierent behavior. Conceptually, this is not a
crucial problem. Sincethe elementary timescale
0
inthe MonteCarloprocedure is arbitrary,
cf. app. D, we can always adjust
0
=
0
(N)in order toreproduce the desiredN dependence.
Apart from the characteristic time scale
d
for translational motion, there are two other
characteristic time scales
S
and
O
in the model, which refer to the change of shape and
orientationof the ellipsoids. Thesecan beidentied by the decay of the respective correlation
0.1 1 10 100 1000
10 100 1000
〈 [r (t)- r(0)] 2 〉
t
~ t
— R 2 G (N)
N=400 N=200 N=100 N=50 N=30
Figure 5.6: Mean square displacement h[r (t) r(0)]
2
i asa functionof time forvariousvaluesof N
in a dense system (c=0:85, =1:0). The linesindicate a linear growth. Forcomparison the mean
in a dense system (c=0:85, =1:0). The linesindicate a linear growth. Forcomparison the mean