Sedimentation path theory for mass-polydisperse colloidal systems

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Tobias Eckert, Matthias Schmidt,∗and Daniel de las Heras†

Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany

(Dated: November 28, 2022)

Both polydispersity and the presence of a gravitational ﬁeld are inherent to essentially any col-

loidal experiment. While several theoretical works have focused on the eﬀect of polydispersity on

the bulk phase behavior of a colloidal system, little is known about the eﬀect of a gravitational

ﬁeld on a polydisperse colloidal suspension. We extend here sedimentation path theory to study

sedimentation-diﬀusion-equilibrium of a mass-polydisperse colloidal system: the particles possess

diﬀerent buoyant masses but they are otherwise identical. The model helps to understand the in-

terplay between gravity and polydispersity on sedimentation experiments. Since the theory can be

applied to any parent distribution of buoyant masses, it can be also used to study sedimentation of

monodisperse colloidal systems. We ﬁnd that mass-polydispersity has a strong inﬂuence in colloidal

systems near density matching for which the bare density of the colloidal particles equals the solvent

density. To illustrate the theory, we study crystallization in sedimentation-diﬀusion-equilibrium of

a suspension of mass-polydisperse hard spheres.

I. INTRODUCTION

A certain degree of polydispersity in e.g. the size and

the shape of the particles is inherent to all natural col-

loids. Even tough modern synthesis techniques allow the

preparation of almost monodisperse colloidal particles [1–

4], a small degree of polydispersity is unavoidable. Un-

derstanding bulk phase equilibria in polydisperse systems

is a signiﬁcant challenge [5]. Polydispersity alters the rel-

ative stability between bulk phases [6–10]. Phases that

are metastable in the corresponding monodisperse sys-

tem can become stable due to polydispersity. Examples

are the occurrence of hexatic columnar [11] and smec-

tic phases [12] in polydisperse discotic liquid crystals,

as well as macrophase separation in diblock copolymer

melts [13]. The opposite phenomenon can also occur. For

example, crystallization in a suspension of hard-spheres is

suppressed above a terminal polydispersity [14–16]. Also,

fractionation into several phases appears if the degree of

polydispersity is high enough [17–19]. A smectic phase of

colloidal rods is no longer stable above a terminal poly-

dispersity in the length of the particles [20]. Dynam-

ical processes such as shear-induced crystallization [21]

are also aﬀected by polydispersity. During drying, a

strong stratiﬁcation occurs in polydisperse colloidal sus-

pensions [22, 23], and the dynamics of large and small

particles is diﬀerent if the colloidal concentration is large

enough [24, 25].

Sedimentation-diﬀusion-equilibrium experiments are a

primary tool to investigate bulk phenomena in colloidal

suspensions. However, the eﬀect of the gravitational

ﬁeld on the suspension is far from trivial [26–30] and

it needs to be understood in order to draw correct con-

clusions about the bulk [31]. Gravity adds another level

of complexity to the already intricate bulk phenomena

∗Matthias.Schmidt@uni-bayreuth.de

†delasheras.daniel@gmail.com; www.danieldelasheras.com

of a polydisperse suspension. To understand the inter-

play between sedimentation and polydispersity, we in-

troduce here a mass-polydisperse colloidal suspension: a

collection of colloidal particles with the same size and

shape (and also identical interparticle interactions) but

with buoyant masses that follow a continuous distribu-

tion. Since the interparticle interactions are identical,

mass-polydispersity does not have any eﬀect in the bulk

phase behaviour. Hence, our model isolates the eﬀects

of a gravitational ﬁeld on a polydisperse colloidal sys-

tem from the eﬀects that shape- and size-polydispersity

generate in bulk.

We formulate a theory for mass-polydisperse colloidal

systems in sedimentation-diﬀusion-equilibrium. The the-

ory is based on sedimentation path theory [32, 33] which

incorporates the eﬀect of gravity on top of the bulk de-

scription of the system. Sedimentation path theory uses

a local equilibrium approximation to describe how the

chemical potential of a sample under gravity changes

with the altitude. So far, sedimentation path theory has

been used to study sedimentation in colloidal binary mix-

tures [28, 31–39]. In this work, we extend sedimentation

path theory to mass-polydisperse systems. Using statis-

tical mechanics, we obtain the exact expression for the

sedimentation path of the mass-polydisperse suspension

combining the individual paths of all particles in the dis-

tribution. We use a model bulk system to illustrate and

highlight the key concepts of the theory, such as the con-

struction of the sedimentation path and that of the stack-

ing diagram (which is the analogue of the bulk phase

diagram in sedimentation). The theory is general and

can be applied to any colloidal system in sedimentation-

diﬀusion-equilibrium. Moreover, the theory contains the

description of a monodisperse system as a special limit

(delta distribution of the buoyant masses). As a proof of

concept, we study sedimentation of a suspension of mass-

polydisperse hard-spheres with diﬀerent buoyant mass

distributions. We ﬁnd that mass polydispersity plays a

major role in systems near density matching. For exam-

ple, near density matching the packing fraction and the

arXiv:2210.04862v2 [cond-mat.soft] 24 Nov 2022

height of the sample at which crystallization is observed

in sedimentation-diﬀusion-equilibrium are strongly inﬂu-

enced by the details of the mass distribution.

II. THEORY

A. Bulk

We use classical statistical mechanics to describe

the thermodynamic bulk equilibrium of our mass-

polydisperse colloidal system. The term bulk refers here

to an inﬁnitely large system in which boundary eﬀects

can be neglected and that is not subject to any external

ﬁeld. The particles diﬀer only in their buoyant masses.

Since the buoyant mass does not play any role in bulk,

the bulk phenomenology of our model is identical to that

of a monocomponent system in which only one buoyant

mass is present. Only when gravity is incorporated into

both systems the buoyant mass becomes a relevant pa-

rameter and the behaviour of the mass-polydisperse and

the monodisperse colloidal systems will diﬀer from each

other.

The total Helmholtz free energy Fis the sum of the

ideal and the excess contributions, i.e. F=Fid +Fexc.

In a mass-polydisperse system, the free energy is a func-

tional of ρm, the density distribution of species with

buoyant mass m. For simplicity, we work with a scaled,

dimensionless, buoyant mass m=mb/m0, where mbis

the actual buoyant mass of a particle and m0is a refer-

ence buoyant mass. Sensible choices relate m0to e.g. the

average buoyant mass of the distribution or its standard

deviation. The concrete deﬁnition of m0is given below

in each considered system.

The ideal contribution to the free energy is a functional

of ρmand is given exactly by

Fid[ρm] = kBTZdm ρm(ln(ρm)−1),(1)

where kBis the Boltzmann’s constant and Tis the ab-

solute temperature. Without loss of generality we mea-

sure ρmrelative to the thermal de Broglie wavelengths

Λm=p2π~2/(mbkBT)with reduced Planck’s constant

~. Note that the value of Λmdoes not play any role

here since altering Λmsimply adds a term to the free en-

ergy that is proportional to the total number of particles

with buoyant mass m. Such term can be reinterpreted

as a change of the origin of the chemical potential of the

species with buoyant mass m.

The integration over min Eq. (1) reﬂects the fact that

due to the mass-polydispersity, the buoyant mass is a

continuous variable. For the shake of simplicity, we omit

the positional argument rin the density distribution

as well as its corresponding space integral that appear

in bulk-phases with positional order such as crystalline

phases.

The ideal free energy, Eq. (1), accounts for the entropy

of mixing of our mass-polydisperse system. The overall

density across all species ρfollows directly from the den-

sity distribution of buoyant masses

ρ=Zdm ρm.(2)

Since the interparticle interaction is independent of the

buoyant masses of the particles, only the density across

all species ρenters into the excess (over ideal) free energy.

Hence, the excess free energy functional must satisfy

Fexc[ρm] = Fexc[ρ].(3)

The grand potential is also a functional of ρmgiven by

Ω[ρm] = Fid[ρm] + Fexc[ρ]−Zdm ρmµm,(4)

where µmis the chemical potential of the species with

buoyant mass m. In equilibrium Ω[ρm]is minimal w.r.t.

the mass-density distribution, i.e.

δΩ[ρm]

δρm0

= 0.(5)

The Euler-Lagrange equation associated to Eq. (5), see

derivation in Appendix A, reads

ln(ρm)−ln(ρ) + βµ −βµm= 0,(6)

where µis the chemical potential of a monodisperse sys-

tem with overall density ρ, see Eq. (2). Hence, it follows

from Eq. (6) that the density of particles with buoyant

mass mcan be written as

ρm=ρeβ(µm−µ).(7)

Integrating Eq. (7) over mon both sides, and us-

ing Eq. (2) on the left hand side, leads to

ρ=ρZdm eβ(µm−µ).(8)

Since ρ6= 0, we obtain

eβµ =Zdm eβµm,(9)

which constitutes an exact analytic expression for the

chemical potential of the monodisperse bulk system

µ=kBTln Zdm eβµm,(10)

in terms of the chemical potentials of the individual

species µmin the mass-polydisperse system. In a

monodisperse system there exists only a single species

and Eq. (10) holds trivially.

B. Particle Model

To proceed we need the bulk equation of state (EOS)

of the monodisperse colloidal system, ρEOS(µ). Given an

interparticle interaction potential, several methods can

be used to obtain the corresponding bulk EOS. These in-

clude e.g. density functional theory [40], liquid state inte-

gral equation theory [41–43], computer simulations [44–

46] and empirical expressions [47–49]. Here, and with the

only purpose of illustrating our theory we use a model

(fabricated) EOS that contains two phase transitions,

see Fig. 1(a). Our model EOS satisﬁes both the ideal

gas limit

lim

µ→−∞ ρEOS(µ)∼eβµ,(11)

and also the close packing limit characteristic of systems

with hard core interactions

lim

µ→∞ ηEOS(µ)/ηcp = 1,(12)

here ηEOS is the packing fraction (percentage of volume

occupied by the particles) according to the EOS and ηcp

is the close packing fraction. Such EOS could repre-

sent e.g. a lyotropic colloidal system with two ﬁrst-order

bulk phase transition, say isotropic-nematic and nematic-

smectic.

Apart from the model EOS, we also illustrate and val-

idate the theory by studying sedimentation of a suspen-

sion of hard-spheres. We use the analytical EOS pro-

posed by Hall [50], which describes the liquid (L) and

solid crystalline (S) phases of a hard sphere system. The

Hall EOS was originally formulated using the compress-

ibility factor as a function of the density. Following

Ref. [51], we numerically integrate the analytical Hall

EOS to obtain the chemical potential as a function of

the density, see Fig. 1(b) for a graphical representation.

It is suﬃcient to ﬁx ρEOS(µ)up to an arbitrary additive

constant in µ. Hence, for convenience, we choose µ= 0

as the chemical potential at the liquid-solid ﬁrst order

phase transition.

C. Sedimentation

To incorporate gravity into our theory, we extend sed-

imentation path theory [32, 52] as formulated for ﬁnite

height samples [31, 33] to include mass-polydispersity. As

often done in colloidal sedimentation, we assume that all

horizontal slices of a sample in sedimentation-diﬀusion-

equilibrium can be described as a bulk equilibrium state,

and also that they are independent of each other. This

local-equilibrium approximation is justiﬁed if the corre-

lation lengths are small compared to the gravitational

lengths ξm=kBT/(mbg), which is the case in many col-

loidal systems. Here gis the acceleration of gravity.

We work in units of the thermal energy kBT, the grav-

itational constant g, and the reference mass m0for ease

Figure 1. Packing fraction ηEOS relative to close packing ηcp

as a function of the scaled chemical potential βµ for (a) our

model equation of state, and (b) the Hall equation of state [50]

for hard spheres. Our model EOS (a) contains three diﬀerent

bulk phases named A,Band Cwhich could correspond to

e.g. the isotropic, the nematic, and the smectic phases of a

lyotropic liquid crystal. The Hall EOS (b) describes the liq-

uid (L) and the solid crystalline (S) phases of a hard-sphere

system. The vertical dotted lines indicate the chemical poten-

tials of the diﬀerent bulk phase transitions. Without loss of

generality, we have translated the origin of chemical potential

such that it coincides with the chemical potential of (a) the

A-B, and (b) the L-Stransitions.

of comparability between diﬀerent systems. Using m0we

deﬁne a reference gravitational length ξ=kBT/(m0g),

which acts as our fundamental length scale.

We treat the slices for each elevation zas a bulk system

with local chemical potentials for each species µmgiven

µm(z) = µ0

m−mbgz, (13)

Here µ0

mis the chemical potential of the species with

buoyant mass mat elevation z= 0. The set of con-

stant oﬀsets µ0

min µm(z)is a priori unknown and must

be determined via an iterative numerical procedure to

match the prescribed mass-resolved density distribution

ρm. Returning to the discussion about the thermal wave-

lengths, altering the value of Λmwould only introduce a

constant term ln(Λm)in Eq. (6) that can be reabsorbed

in Eq. (13) as a shift of the chemical potential µmvia

the oﬀset µ0

m. The oﬀsets µ0

mdepend therefore on the

choice of Λm. However, the sedimentation proﬁles ρm(z)

remain unchanged, since µ0

mare determined to match the

prescribed density distribution.

Equation (13) is the sedimentation path [31–33, 52] of

the species with buoyant mass m. It hence describes how

the chemical potential of each species varies linearly with

zin the range 0≤z≤h, with hthe sample height. The

local chemical potential for each species either decreases

(mb>0) or increases (mb<0) with the elevation z,

depending on the sign of the buoyant mass.

The sedimentation path of each species µm(z)is just a

straight line, see Fig. 2(a), as in the case of monodisperse

systems. Next, we combine all paths at each elevation z

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Sedimentationpaththeoryformass-polydispersecolloidalsystemsTobiasEckert,MatthiasSchmidt,andDanieldelasHerasyTheoretischePhysikII,PhysikalischesInstitut,UniversitätBayreuth,D-95440Bayreuth,Germany(Dated:November28,2022)Bothpolydispersityandthepresenceofagravitationaleldareinherenttoessentiallyanyco...

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Sedimentation path theory for mass-polydisperse colloidal systems

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