Journal Name

Dynamic Phase Diagram of Soft NanocolloidsSupplemental Material

Sudipta Gupta,∗a,b Manuel Camargo,c Jörg Stellbrink,a Jürgen Allgaier,a

Aurel Radulescu,d Peter Lindner,e Emanuela Zaccarelli, f Christos N. Likos,g

and Dieter Richtera

SANS modelingThe small angle neutron scattering (SANS) scattering cross sec-tion dΣ

dΩ(Q) in absolute units [cm−1] following core-shell model1–3

in dilute solution is given by (cf. Eq. (3) of the main paper)(dΣ

dΩ

)(Q) = Nz Ics(Q)

= Nz

[Icore(Q)+ Ib

corona(Q)+ Iinter(Q)+ Iblob(Q)]

=φ

Vm

[V 2

coreN2agg∆ρ

2coreA2

core

+V 2coronaNagg

(Nagg−

11+ ν

)∆ρ

2coronaA2

corona

+2VcoreVcoronaN2agg∆ρcore∆ρcoronaAcoreAcorona

+V 2coronaNagg∆ρ

2corona

(Pp(Q)

1+ νPp(Q)

)](1)

where ∆ρcore = ρcore− ρsolvent and ∆ρcorona = ρcorona− ρsolvent

are the contrast difference of the core and corona of the micelleswith respect to the solvent and ρi the corresponding scatteringlength densities. Vcore and Vcorona are the volume per molecule

a JCNS-1 and ICS-1, Forschungszentrum Jülich, Leo-Brandt-Straße, 52425 Jülich, Ger-many. E-mail: s.gupta@fz-juelich.de, j.stellbrink@fz-juelich.deb JCNS-SNS, Oak Ridge National Laboratory (ORNL), Bethel Valley Road, TN-37831Oak Ridge, USA.c Centro de Investigaciones en Ciencias Básicas y Aplicadas, Universidad Antonio Nari-ño, Km 18 via Cali-Jamundí, 760030 Santiago de Cali, Colombia.d JCNS-FRM II, Forschungszentrum Garching, Lichtenbergstraße 1, 85747 Garching,Germany.e Institute Laue-Langevin, 6, rue Jules Horowitz, 38042 Grenoble CEDEX 9, France.f CNR-ISC and Dipartimento di Fisica, Universitá di Roma La Sapienza Piazzale A.Moro 2, I-00185, Roma, Italy.g Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria.

of the insoluble core and soluble corona blocks, respectively de-fined as Vi = Mi/(diNA), with NA the Avogadro’s number, di thebulk density and Mi the molecular weight in g/mol of the core orcorona blocks. Vm =Nagg (Vcore +Vcorona) as the micellar volume. ν

is an effective virial type excluded volume parameter that scaleswith the effective concentration of the corona chains. Follow-ing Svaneborg and Pedersen,2,4 the blob scattering from swollencorona chains was modeled as Iblob(Q). In this model2 the chainsare considered to be self-avoiding and interact mutually by blobsand also with the homogeneous core following a hard sphere po-tential. The scattering amplitude from the core is given by

Acore(Q) =

Rc∫0

dr4πr2 sin(Qr)Qr ϕcore(r)

Rc∫0

dr4πr2ϕcore(r)exp(−c2

s Q2

2

)

= 3sin(QRc)− (QRc)cos(QRc)

(QRc)3 exp

(−c2

s Q2

2

)(2)

for a core density profile ϕcore=1 (compact core). For strongsegregation the core smearing parameter is generally kept cs=0,causing no effective change in the scattering pattern.

The scattering amplitude from the corona or shell is given by

Acorona(Q) =

Rm∫Rc

dr4πr2 sin(Qr)Qr ϕstar(r)

Rm∫Rc

dr4πr2ϕstar(r)exp(− s2

s Q2

2

)(3)

Where ϕstar(r) the star-like density profile. For a finite size of thecorona, the limit is chosen to be the micellar radius Rm, instead of∞, with ss the smearing parameter for the corona. For the linearchain form factor one can use the Beaucage form factor5,6 given

Journal Name, [year], [vol.], 1–2 | 1

Electronic Supplementary Material (ESI) for Nanoscale.This journal is © The Royal Society of Chemistry 2015

by:

Pp(Q) = exp

(−

Q2R2g

3

)+

d f

Rd fg

Γ

(d f

2

)er f(

RgQk√

6)3

Q

d f

(4)

Where, k = 1.06, Γ is the Gamma function and d f is the fractaldimension of the scattering particle, 1≤ d f ≤ 3. In this approacha polymer chain is considered as a mass fractal characterized by asingle spatial length scale, the radius of gyration Rg of the linearchain, and a fractal dimension for polymers in good solvent istypically d f = 1.7. The first term in Eq.(4) is from the Guinierexpression7.

DLS and rheology modelingFor dynamic light scattering (DLS) our data analysis of the ex-perimental intensity auto correlation function (IACF) g(2)e (Q, t)was based on the inverse-Laplace transformation by CONTIN al-gorithm developed by Provencher8,9 i.e.

g(1)e (Q, t) =1

2π

∞∫0

dΓG(Γ)exp(−Γt) (5)

where Γ = DQ2, with D the diffusion coeffecient of the scatteringparticles. We use the CONTIN algorithm as provided by the ALV-software.

To yield the zero-shear viscosity η0 and to investigate theshear rate dependent viscosity (shear thinning) the Carreau equa-tion10 is used, which is given by:

η(γ)−η∞

η0−η∞

=1

[1+(γ/γc)a]1−b

a

(6)

Where η∞ denotes high shear rate Newtonian limit of viscosity.Frequently the high shear rate region is not observed, and η∞ isset to zero in Eq.(6). γc indicates the onset of the shear thinningand has the dimensions of s−1; the power law exponent, (1− b),describes the dependence of the viscosity on shear rate in theshear thinning region. For our samples the value of (1− b) liesbetween 0.2 to 0.76 at intermediate concentration for φ < φ∗m. Itis to be noted that, for dilute concentration (1−b) = 0, gives theNewtonian plateau with zero-shear viscosity η0. The additionaldimensionless parameter ’a’ represents the width of the transitionregion between the constant Newtonian plateau observed at lowshear rates and the asymptotic power law decrease of the viscosityfound at high shear rates. Value of a = 2 is kept constant.

The Krieger-Dougherty (KD) model for solutions of sphericalsuspensions, it is given by :

η0(φ)

ηsolv=

(1−

φe f f

φlim

)−ε

(KD) (7)

Here φe f f = φ , the effective volume fraction and ε = [η ]× φlim,and Martin relations for solutions of spherical suspensions

η0(φ)

ηsolv= 1+[η ]φe[η ]Kφ (Martin relation) (8)

for φ > φ∗. Here, K is a constant and [η ] = (η0−ηsolv)/(ηsolvφ) is

the intrinsic viscosity in the limit φ → 0, of the system11.

References1 S. Gupta, PhD Thesis, University of Münster, 2012.2 J. S. Pedersen, C. Svaneborg, K. Almdal, I. W. Hamley and

R. N. Young, Macromolecules, 2003, 36, 416–433.3 R. Lund, V. Pipich, L. Willner, A. Radulescu, J. Colmenero and

D. Richter, Soft Matter, 2011, 7, 1491–1500.4 C. Svaneborg and J. S. Pedersen, Phys. Rev. E, 2001, 64,

010802.5 G. Beaucage, J. Appl. Cryst, 1995, 28, 717.6 G. Beaucage, J. Appl. Cryst, 1996, 29, 134.7 P. Linder, Neutrons, X-rays, and Light: Scattering methods ap-

plied to soft condensed matter, Elsevier, North-Holland DeltaSeries, Amsterdam, 2002.

8 S. W. Provencher, Computer Physics Communications, 1982,27, 229–242.

9 S. W. Provencher, Computer Physics Communications, 1982,27, 213–227.

10 R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Poly-meric Liquids, John Wiley and Sons, New york, 1987, vol. 1.

11 W. M. Macosko, Rheology Principles, Measurements and Appli-cations, Wiley-VCH, New York, 1994.

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