Polymer Shape Anisotropy and the Depletion Interaction

4 Pages·0.14 MB·English

by  Mario Triantafillou

Polymer Shape Anisotropy and the Depletion Interaction
Mario Triantafillou and Randall D. Kamien
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
(19 October1998; revised 7 December1998)
We calculate the second and third virial coefficients of the effective sphere-sphere interaction due
topolymer-induceddepletionforces. Byutilizingtheanisotropyofatypicalpolymerconformation,
9 we can consider polymers that are roughly the same size as the spheres. We argue that recent
9 experiments are laboratory evidencefor polymershape anisotropy.
9
1 PACS numbers: 61.25.Hq, 82.70.Dd, 36.20.Ey

n
Over sixty years ago,Kuhn studied the conformations spheres are excluded. When two spheres are close their
a
J of polymer chains [1] and recognized that typical con- excludedregionsoverlapleavingmorefreevolumeforthe
formationsof idealchains arenotspherically symmetric. remaining spheres and hence a larger entropy. Thus the
6
2 The intuitive idea ofasymmetric shape is aresultofthe propensity for “close” configurations can be interpreted
isotropicend-to-endvectordistributionofarandomwalk as an entropic “depletion” force.

] [2]: spherical symmetry results from implicit rotational Recently [9], monodisperse polymers (specifically, λ-
t
f averagingofthepolymer. Infact,atypicalpolymercon- phage DNA) have been used to induce depletion forces
o
formation is anisotropic, with an aspect ratio of roughly between polystyrene spheres. To model this system at
s
. 3.4:1 [3]. low concentrations,one mightreplace the polymers with
t
a Nonetheless, there is little laboratoryevidence for this spheres of radius RG, the polymer radius of gyration.
m asphericity. In solution, polymers rotate randomly and Asakura and Oosawa derived a simple formula for the
- there are no significant polymer-polymer correlations. potentialbetweentwolargespheresofdiameterσinagas
d Thus small angle light scattering measurements of poly- of smaller spheres of diameter D. The Asakura–Oosawa
n
mer solutions can only determine the average princi- (AO) potential is [8]:
o
pal axis of the polymer shape. If there were a way to
[c induce strong orientational correlations, light-scattering U(R) Πvλ3 3 R 1 R 3
2 msheoawsutrheamtetnhtesincoduulcdeddaistcterranctiaonn baseytwmemenetrreyg.ionWseofwdiel-l kBT = (λ−1)3 "1− 2(cid:18)σλ(cid:19)+ 2(cid:18)σλ(cid:19) #, (3)
v
4 pleted polymer concentration (inclusions) is a probe of where R is the distance between the centers of the large
3 this shape anisotropy. spheres, v is the volume of a small sphere, Π is the os-
2 Theshapetensorcharacterizesthespatialdistribution motic pressureof the small-sphere gasand λ 1+D/σ.
0 of monomers: The above approximation simply sets D =2R≡ . This is
G
1
N obviously a crude approximation to the true system. A
8 M = dn R (n) R¯ R (n) R¯ , (1) complete analysis at length-scales longer than the poly-
/9 αβ Z0 α − α β − β mer persistence length (50nm) would count the num-
t (cid:0) (cid:1)(cid:0) (cid:1) ber of self-avoiding random walks which avoid the two
a whereR (n)isthepositionofmonomern,αandβ label
α
m the Cartesian co¨ordinates and R¯ 1 NdnR (n) is polystyrene spheres.
- the polymer center of mass. Theαp≡olyNmer0radiusαof gy- In general,the effective potential is of the form Ueff =
d rationis simply R2 =Tr(M). Moreover,Rthe eigenvalues ΠV(R) where Π is the osmotic pressure and V is an
n G R dependent recovered volume. At low concentrations
o of Mαβ, λ21 ≤ λ22 ≤ λ23, are the average squared radii of the osmotic pressure is not adjustable: Π = k Tc. The
c gyration along the principal axes of inertia. Simulations B
physics all lies in V(R). The AO model gives a one-
: [3,4] have determined that the the most likely shape has
v parameterfamilyoffunctions,dependingontheeffective
Xi λ2 :λ2 :λ2 11.8:2.7:1.0 (2) hard-sphere diameter D. In principle one could derive a
3 2 1
≈ virialexpansionfor this potentialwith eachterm involv-
r
a Indeed, the shape asymmetry exists and is rather large ing the evaluation of a set of cluster integrals, each of
[5]. Exploiting this shape anisotropy as a calculational which involves integrations overpolymer degrees of free-
tool is the main theoretical component of this letter. dom. We will derive a different one-parameter family
Though hard spheres only interact by direct contact, based on the known shape distribution of polymers and
entropiceffectsofotherparticlespresentcaninducelong- arguethat the data in [9] is the first laboratoryevidence
range interactions. These sorts of forces are responsible of the conformational anisotropy.
for liquid-crystalline order in lyotropic systems [6] and We start by calculating the classical configurational
surface crystallization in hard-sphere fluids [7]. It is in- integral Q (R), the sum of Boltzmann weights over all
N
structive to consider the virial expansion for a gas of conformations of N polymers with two spherical inclu-
identical balls. Aroundeachsphere of radiusr there is a sionsseparatedbyR. The sphere-sphereeffective poten-
sphere of radius 2r from which the centers of the other tial Ueff(R) is:
1
QN(R) exp βUeff(R) than a sphere. More importantly, our approximation al-
P(R)= {− } . (4)
dRQN(R) ≡ dRexp βUeff(R) lowsusto considerpolymerswhichareroughlythe same
{− }
size as the included spheres. This has a great advan-
Although QRN(R) has termsRwhichare independent of R, tage when comparing to the experiments of Verma, et al
the resulting effective potential Ueff(R) does not: they [9]. By comparison, most work in this field has treated
cancel between numerator and denominator in (4). the inclusions approximately while correctly modelling
TocalculateQN(R),wesumoveralltheconformations the medium as a gas of random walkers. In [10] field
andplacementsofN polymerswiththeexcluded-volume theoretic methods were used to obtain the depletion po-
Boltzmannweights: 1ifthepolymersandthespheresdo tential. There, in order to calculate reliably, the au-
notoverlapand0otherwise. Wesplittheintegrationover thorsconsideredtheinteractionbetweeninclusionsmuch
eachpolymerintothreeparts. Thefirstisanintegration smaller than the polymers, i.e. σ R . In the
G
over the center of mass, the second an integration over opposite extreme, one might consider≪polymers which
all rigid rotations of the conformation, while the third is are much smaller than the spheres. In this case, it
the remainingintegrationover“internal”degreesoffree- is appropriate to study the induced Casimir force be-
dom. This final integral is over each unique conforma- tweentwowallsand,tothenapproximatetheinteraction
tion – two conformations are equivalent if one is merely between spheres via the Derjaguin approximation [11].
arigidrotationortranslationsofthe other. Wewillonly By approximating the polymers as prolate spheroids,
integrate over one representative from each equivalency we can treat the inclusions exactly – a great advan-
class. Denoting the space of all such “internal” polymer tage when the two extreme limits are not applicable.
conformations by Υ, we have:
N
1
QN = dri,int dridΩie−βU, (5)
N!i=1ZΥ Z
Y
where dri,int is the measure on the space of conforma-
tions for polymer i, r is its center of mass and Ω is
i i
its rigid rotation. We further divide the integration over
Υ by characterizing each polymer conformation by its
principal axes. Defining g(λ1,λ2,λ3) is the number of
(a) (b) (c)
conformations with axes λ , we have
i
FIG.1. Sequence of approximations. The actual polymer
N
1 performsaself-avoiding randomwalk (a)whichhasatypical
Q =
N N! prolate spheroidal shape (b). For computational simplicity,
iY=1 we neglect the anisotropy between the two shorter principal
dλi1dλi2dλi3g(λi1,λi2,λi3) dridΩie−βU. (6) axes and build up the resulting ellipsoid out of overlapping
spheres (c).
Z Z
To pass from(5) to (6)we assumedthat the internalde-
We have thus reduced the configurationalintegral Q
grees of freedom did not affect the interaction potential N
over all polymer modes to an integral over the allowed
U. This is, of course, not precisely correct. Our approx-
locationsandorientationsofaprolatespheroid. Inorder
imation replaces each polymer by a solid ellipsoid, and
toreducethephasespacesomewhatandreduceourcom-
then considers the potential due to only to this shape.
putation time, we will assume that λ1 =λ2. In the case
Whilethis certainlyremovesmanydegreesoffreedom,it
of interest, we consider a prolate spheroid with aspect
includesmoredegreesoffreedomthanreplacingthepoly-
ratio √11.8:1 3.4:1.0, so that
mers by spheres. The proof of the pudding shall be in ≈
the eating – we will see that this approximation is valid g(λ1,λ2,λ3)
by comparison with data [9]. Thus our approximation ∝ 2 2 2
replaces the monomer–sphere and monomer–monomer δ(λ1−λ2)δ λ3−(3.4)λ1 δ RG−[2λ1+λ3] , (7)
potential with a sum of pairwise, ellipsoid-ellipsoid or where R is the (cid:0)radius of gyra(cid:1)ti(cid:0)on of the polyme(cid:1)r. To
G
sphere-ellipsoidterms. Eachtermisinfiniteforanyover- facilitate the numerical evaluation of integrals, we con-
lap and zero otherwise. struct the ellipsoid out of overlapping spheres. Figure
We now reduce the complexity of the integration in 1 depicts our sequence of approximations. We note that
(6) by using the shape distribution of polymers. Since choosingλ3 λ2 =λ1 wouldreduceouranalysistothat
≫
the distribution of polymer shapes is peaked around in [12]which consideredthe depletion force between two
the prolate spheroid, we only consider those polymer spheres induced by thin rods.
shapes. This approximation does not account for the Writing fij = e−βuij 1, where uij is the excluded-
−
entire space of principal axes, though we believe that volumepotentialbetweenparticlesiandj,andlabelling
it does characterize the polymer conformations better the spheres A and B, Q (R) can be rewritten as
N
2
1
QN= e−βuAB dr1dΩ1dr2dΩ2...drNdΩN (8)
N! × 0.1
ZV
(1+fA1)(1+fB1)(1+fA2)(1+fB2)(1+f12)... 0
T
Theproductin8isasumoftermswhichcanbegrouped B −0.1
by the number of polymer positions and rotations that )/k c = 0.2/µm3
are freely integrated over. The first term is proportional (R −0.2 c = 0.5/µm3
to(4πV)N,theconfigurationalintegralforfreeellipsoids, Ueff −0.3 c = 1.0/µm3
where V is the volume of space minus the volume of the c = 2.2/µm3
−0.4
twoincludedspheres. Subsequenttermshavefewerpow-
1.2 1.4 1.6 1.8 2 2.2
ers of V. Since the polymers are identical, these correc- R [µm]
tions include combinatoric factors involving N. We take
thelimitN andV keepingc N/V constant
to find the v→iria∞l expansi→on∞in c. ≡ FIG.2. NumericalresultfortheeffectivepotentialUeff for
differing concentrations and RG = 0.5µm. We have kept
Itisbothconvenientandinstructivetographicallyrep-
termsuptosecondorderinthepolymerconcentration. Note
resent these “cluster” integrals [13] in terms of Mayer
that at the highest concentration there is a small repulsive
cluster graphs. The first two terms in Ueff(R) are: bump at R ≈2µm. However, for c≤c∗ ≈2/µm3, the third
c q 1 c 2 virial coefficient is a small contribution to Ueff.
βUeff(R)= dr1dΩ1 a a + (9)
4π 2 4π
dr1dr2dΩZ1dΩ2 2aq aq+2aq aq+(cid:16)2aq (cid:17)aq+2aq aq+ aq aq moWdeelcaotmcpa=re0o.u5r/µmmod3eflowrittwhothreeadsaotnas.[9]T∗ahnids cthonecAenO-
× tration is below the overlap concentration c and at the
Z (cid:20) (cid:21)
same time is large enough that the well depth is on the
where the open dots represent the spherical inclusions order of k T so that data can be reliably obtained. We
B
and the closed dots represent the ellipsoids. The inte- canadjusttheeffectiveradiusR forboththeAOmodel
G
gralsin(9)aredifficulttocomputeanalyticallyandthus and our ellipsoid-based model. We find that a good fit
wereevaluatednumericallyviaaMonteCarloalgorithm: results for the AO model with R =0.42µmand for our
G
103 different angles and 104 different points were chosen model with R = 0.8µm, in comparison with the light-
G
in a volume which included both spheres and which did scattering-obtained value R = 0.5µm. In Figure 3 we
G
not exclude any possible orientation or location of the show the data along with these two one-parameter fits.
ellipsoids. We calculated these integrals by this random We have checked other concentrations and have found
sampling weighted by the appropriate phase-space vol- that at c = 1.0/µm3 the theory and experiment also
ume factor. compare favorably with the same effective radii. What
To compare with experiment [9], we took the sphere should one conclude from this agreement? It is possible
diameter to be D = 1.2µm and RG = 0.5µm. Know- to interpret the data as arising from either shape simply
ing RG enables us to find the length of the ellipsoid, by adjusting the size of the shape. However, one should
L = 3.4 2RG/√13.8 0.92µm. We have chosen the start from a microscopic picture based on the polymer
× ≈
hard-ellipsoid radii to be the mean square radii of gy- physics of the allowed chain conformations. Only from
ration, which is possibly na¨ıve: the hard-ellipsoid size this perspective can one properly interpret the data.
should only be proportional to RG. Since, in a random Since the radius of gyration of the λ-DNA can be cal-
walkthe density decaysas1/r there is no naturallength culated fromits molecular weightand persistence length
scale whichcuts offthe excluded-volumeinteraction. In- to be R = 0.5µm we can consider two different zero
G
deed, light scattering experiments [14] have found that parameter fits: our model and the AO model. The re-
the effective hard sphere radius is roughly half the ra- sultisshowninFigure4. Note thatthe dataliebetween
diusofgyration. TherelationbetweenRG andthehard- our calculation and the AO model, suggesting that one
ellipsoid size must be determined through the depletion- could smoothly deform the AO sphere into an ellipsoid
force experiments we are modeling. and find a best fit aspect ratio to fit the data, giving
In Figure 2 we plot our results as a function of con- a one-parameter fit. Indeed, we have made a number
centration. Until one considers concentrations near the of limited runs on the fully anisotropic shape satisfy-
p2/oµlymm3erthoevetrhlaiprdcovnircieanltrcaoteiffionciecn∗t,=res1p/o(n4sπiRblG3e/3fo)r ≈a idnegep(e2r).thTanhitshraetsuinltsFiinguarec4u,rvwehwichhicihs bisetrtoeurgthhlayn3t0h%e
repulsive “anti-correlation hole”, is a small perturba- simple sphero-cylinder result. Finally, we have made
tion to the leading term in (9). Thus if we re- similar comparisons between theory and experiment at
strict our study to the dilute polymer regime, the c = 0.1/µm3, c = 0.2/µm3 and c = 1.0/µm3. We find
leading term in the virial expansion is sufficient. that at c = 1.0/µm3 the comparison is similar to that
shown in Figure 4. At the two lower concentrations,
where the data is difficult to collect, neither our model
3
nor the AO model make very good predictions. mationsbytreatingthemasrigidellipsoidsisappropriate
forstaticproperties,itisnotclearatallthatthedynam-
ics reflects this. In particular one might ask whether a
0.1 polymer ellipsoid rotates to a new orientation slower or
faster than it deforms into that orientation. Finally, our
analysis could also be used to study the depletion inter-
−0.1 actionbyactualellipsoidalobjects,suchasbacteria[15].
T It is a pleasure to acknowledge conversations with
B
k
R)/ −0.3 J.Crocker,D. Discher,A.Levine,T.Lubensky,D.Pine,
U(eff DAOata Rth.oVrseormf[a9]afonrdpAro.vYidoidnhg.usWweitahdtdhiteiiorndaaltlya.tThahniskrtehseeaarcuh-
Ellipsoid was supported in part by an award from Research Cor-
−0.5
poration, the Donors of The Petroleum Research Fund,
administeredby the AmericanChemical Society andthe
NSF-MRSEC Programthrough Grant DMR96-32598.
−0.7
1.1 1.3 1.5 1.7 1.9 2.1
R [µm]
FIG.3. OurmodelandtheAOmodelarefittotheexper-
imental data for c = 0.5/µm3. We varied only the effective
RG as a parameter. The fit gives RG = 0.42 and 0.8 for the
AO model and our model respectively. [1] W. Kuhn,Kolloid Z. 68, 2 (1934).
[2] M. Doi and S.F. Edwards, The Theory of Polymer Dy-
namics, p. 29, (Oxford UniversityPress, Oxford, 1986).
[3] K. Sˇolc, J. Chem. Phys. 55, 335 (1971).
0.1 [4] D.E. Kranbuehl and P.H. Verdier, J. Chem. Phys. 67,
361. (1977).
−0.1 [5] We note that although polymer self-avoidance leads to
T
B a swelling of the polymer chain, it does not significantly
k
)/ alter the shape anisotropy. See J.A. Aronovitz and D.R.
R −0.3 Nelson, J. Phys. (Paris) 47, 1445 (1986).
(
Ueff Data [6] M. Adams, Z. Dogic, S.L. Keller and S. Fraden, Nature
−0.5 AO model 393, 349 (1998).
Ellipsoid [7] A.D. Dinsmore, A.G. Yodh, D.J. Pine, Nature 383, 239
Average (1996).
−0.7 [8] S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255
1.1 1.3 1.5 1.7 1.9 2.1
(1954)
R [µm]
[9] R. Verma, J. Crocker, T.C. Lubensky and A.G. Yodh,
Phys. Rev.Lett., 81, 404 (1998).
FIG.4. ComparisonoftheAOmodel,themodelpresented [10] A.Hanke,E.EisenrieglerandS.Dietrich,preprint[cond-
hereandthedataatc=0.5/µm3. Forbothourmodelandthe mat/9808225].
AOmodelwehavetakenthetheoreticalvalueofRG ≈0.5µm. [11] J.F. Joanny, L. Leibler and P.-G. de Gennes, J. Polym.
Therearenofreeparametersinthemodels. Wealsoplotthe Sci.,Polym.Phys.Ed.17,1073(1979); seealsoY.Mao,
average of theAO model and theellipsoid calculation. M.E. Cates, and H.N.W. Lekkerkerker, J. Chem. Phys.
106, 3721 (1997).
Returningtotheoriginalquestion,arepolymershapes [12] K. Yaman, C. Jeppesen, and C.M. Marques, Europhys.
anisotropic? Though the full distribution g(λ1,λ2,λ3) is Lett. 42, 221 (1998).
peaked, it has a finite width. We can incorporate this [13] See, for instance, R.K. Pathria, Statistical Mechanics,
width by choosing an appropriately weighted admixture (Butterworth-Heinemann, New York,1996).
of shapes. Note that we could easily get a very good fit [14] X. Ye, T. Narayanan, P. Tong, and J.S. Huang, Phys.
Rev. Lett.76, 4640 (1996)
just by mixing spheres andellipsoids – the mixedsecond
[15] We thankD. Discher for discussions on this point.
virial coefficient is just a linear combination of the two
“pure” coefficients. It is easy to see from Figure 4 that
a 50-50admixture of spheres and ellipsoids would give a
remarkably good fit to the data, without any adjustable
parameters. While we have no basis for this weighting
of shapes, we nonetheless take this as directevidence for
the anisotropy of polymer conformations.
Inclosing,wenotethatwhilecountingpolymerconfor-
4