Scuola Normale Superiore di Pisa
PhD Thesis in Physics
Academic Years 2008–2010
Fluid-fl uid dem ixing cur ves in m ixt ur es of
colloids and p olym er s wit h r andom im pur it ies
A dvisor s: P hD Candidat e
Prof. Andrea Pelissetto Dr. Mario Alberto Annunziata Prof. Ettore Vicari
SN S R efer ence P r of. Prof. Riccardo Barbieri
Cont ent s
1 Exp er im ent s on m ixt ur es of colloids and p olym er s 5
1.1 Colloids as hard spheres . . . 5
1.2 Polymer solutions . . . 9
1.2.1 Dilute regime . . . 11
1.2.2 Semi-dilute regime . . . 14
1.3 Mixtures of colloids and non-adsorbing polymers . . . 16
1.3.1 Colloid limit . . . 16
1.3.2 Protein limit . . . 20
2 T he A OV m odel 24 2.1 The model . . . 24
2.2 Depletion-induced phase separation . . . 25
2.3 Free Volume Theory . . . 30
3 Simulat ion M et hod 36 3.1 Grand Canonical Monte Carlo . . . 36
3.2 Simulated Tempering . . . 42
3.2.1 Simulated tempering . . . 42
3.3 Umbrella sampling . . . 43
4 T he A OV m odel wit h disor der 48 4.1 Introduction . . . 48
4.2 Definitions . . . 49
4.3 The effective volume fraction f . . . 51
4.4 Estimate of f∗ . . . 55
4.5 Simulation results . . . 58
4.5.2 Study of the AOV model in the presence of a colloidal
matrix . . . 72
4.6 Conclusions . . . 81
5 Coar se-gr ained m odel wit h int er act ing p olym er s in t he bulk 83 5.1 Introduction . . . 83
5.2 Definitions . . . 84
5.3 Simulation results . . . 85
5.3.1 Demixing curve in the system representation . . . 91
5.3.2 Demixing curve in the reservoir representation . . . 91
5.3.3 Cumulant ratios . . . 91
5.4 Conclusions . . . 96
6 Coar segr ained m odel wit h int er act ing p olym er s and disor -der 97 6.1 Definitions . . . 97
6.2 Simulation results . . . 98
6.2.1 Finite-size effects . . . 99
6.2.2 Demixing curves in the reservoir representation . . . . 104
6.2.3 Demixing curves in the system representation . . . 107
6.2.4 Connected susceptibility . . . 107
6.3 Conclusions . . . 112
Conclusions 114
P hD act ivit y sum m ar y 116
A cknowledgem ent s 117
Chapt er 1
Exp er im ent s on m ixt ur es of
colloids and p olym er s
In this chapter we review some important experiments done on mixtures of colloids and polymers whose results justify the approximations we used in our simulation work.
1.1
Colloids as har d spher es
In this thesis we consider binary mixtures of polymers and colloids. Several types of colloids have been synthesized. We only discuss uncharged colloids which behave quite precisely as hard spheres. Typical examples are spheres of polymethylmetachrylate (PMMA) with a surface layer of grafted polymers. This layer is necessary to stabilise the colloids, i.e. to prevent them to coalesce because of the very strong van der Waals-like superficial interactions. The corresponding energy is orders of magnitude bigger than the factor kBT at
room temperature, hence stabilization—one way of realizing it is by coating colloids with a thin layer of grafted polymers—is necessary. The addition of a layer of solvophilic polymers creates a repulsive interaction among colloids because a strong repulsion arises when their “coats” intersect.
Stabilised uncharged colloids behave quite precisely as hard spheres. This is shown very clearly by an experiment by Pusey and van Megen (1986) [57]. They considered PMMA spheres of radius Rc = 300 nm stabilised by grafted
polymers that extend over δ ≈ 10 ÷ 20 nm. Spheres of PMMA were put in ten vessels (see Fig. 1.1) in different concentrations. We characterise it by
Figure 1.1: Pusey-van Megen experiment: vessels in order (from right to left) of increasing ηc [57].
using the PMMA (or core) volume fraction
ηc = 4π 3 R 3 c NPMMA V , (1.1)
where NPMMAis the number of PMMA spheres and V is the volume. The
ves-sels were extensively shaken in order to re-disperse the particles in a random fashion. They found (see Fig. 1.1):
• in vessel #2 (the first from the right in Fig. 1.1), the less concentrated one (ηc = 0.393), no spontaneous crystalline structure appeared: the
system acted like a liquid, each colloid behaving as a Brownian particle; • in vessels #3 – #5 (ηc increases) a crystal phase appears, in equilibrium
with a fluid phase;
• in vessels #6 and #7 a homogeneous crystal phase appears throughout the entire vessel;
Figure 1.2: Phase diagram of a mixture of stabilised colloids [57]. The volume fraction is indicated with φ.
• in vessel #9 a heterogeneous crystal phase coexists with a non-crystalline amorphous phase;
• in vessel #10, the most concentrated one (ηc ∼ 0.50), an amorphous
non-crystalline phase is present throughout the entire vessel. These results show the presence of different phases (see Fig. 1.2):
• a fluid phase (ηc < ηFc ≃ 0.407);
• a coexisting fluid-crystal phase (ηF
c < ηc < ηcM ≃ 0.441);
• a homogeneous crystal phase (ηM
c < ηc < ηcC ≃ 0.485);
• a heterogeneous crystal phase (ηC
c < ηc < ηGc ≃ 0.502);
Figure 1.3: Phase diagram of a system of hard spheres (φ is the volume fraction) [22].
Here ηF
c , ηMc , ηCc , ηcG are respectively the freezing, the melting, the
crystal-lizing and the glassy values of the colloid volume fraction.
The growth of the crystal for ηc > 0.4 takes place because of the diffusion
of the particles. However, if ηc is too high, particles are no longer able to
diffuse because they are hindered by their neighbours. Hence a disordered “jammed” phase comes out, with a higher volume fraction but with short-range, liquid-like order. This phase is also called a glassy phase. The presence of a glassy phase is very interesting, also because of its possible relevance for manufacturing high-strength ceramics [57].
The phase diagram reported in Fig. 1.2 is qualitatively the same as that of a system of hard spheres, see Fig. 1.3. There are, however, quantitative differences: for instance, the core volume fraction at which freezing starts, ηF
c ≃ 0.407 [57], is different from the corresponding value for a mixture of
hard spheres, ηF
HS = 0.494. To make a quantitative correspondence we define
an effective radius RHS= Rc+ ǫ, an effective colloid volume fraction
ηeff= 4π 3 R 3 HS NPMMA V (1.2)
is indeed the length of the grafted polymers. Hence a mixture of colloids of radius Rc stabilised with a layer of grafted polymers of thickness δ behaves
like a mixture of hard spheres of radius RHS= Rc+ δ.
With this prescription we obtain for the melting volume fraction ηM
eff = 0.536,
which is only slightly different from the hard-sphere melting volume fraction (ηM
HS = 0.545). This tiny difference could be due to the fact that the
in-teraction potential is not strictly a hard-sphere potential, but rather a soft potential. Moreover, the effective volume fraction of the most concentrated glassy sample, #10, is close to the value ηG
HS= 0.637 expected for the random
close-packed hard-sphere glass.
1.2
Polym er solut ions
Polymers are long molecules obtained by means of a polymerization process that replicates a basic unit many times (more precisely one should speak of homopolymers, the only type of polymers we shall consider). Polymers in a solvent are binary mixtures which have a very simple temperature depen-dence. For each solvent and polymer type there exists a critical temperature Tθ (θ-temperature) which divides the good-solvent (T > Tθ) from the
poor-solvent regime (T < Tθ). For T = Tθ one speaks of θ-polymers.
The presence of these different regimes is due to two competing interactions between the polymer chains and the solvent. The first one is a steric repulsion between the monomers, the segments which constitute the polymer chain: it forces the polymer chain to expand. The second one is a solvent-induced attraction between the segments of the chain which tends to shrink the chain. These two forces depend on temperature and they balance each other at T = Tθ. In a good solvent the steric repulsion prevails, hence the polymer
chain expands. In a poor solvent, instead, the attraction between polymer chains prevails, and dilute polymer chains collapse into globules.
The size of a polymer is usually characterized by the radius of gyration, which is defined as follows. If {~r1,· · · , ~rN} are the positions of the monomers of a
single polymer, we define R2 G as R2G = 1 2N2 X ij (~ri− ~rj)2, (1.3)
For very long chains the radius of gyration has a scaling behaviour of the form
RG∼ Nν, (1.4)
where N is the number of monomers, ν an exponent that depends only on the solvent conditions. In the poor-solvent case
ν = 1
3; (1.5)
the polymer collapses to a globule.
At the θ-point there is effectively no interaction among the units and the system behaves effectively as a random walk (with logarithmic corrections) and
ν = 1
2. (1.6)
In the good-solvent regime the chain is swollen and ν is larger than 1/2. Numerical calculations give1
ν ≈ 0.588, (1.7)
which is close to the approximate Flory expression
ν = 3
5. (1.8)
For T larger than, but still close to, Tθ crossover effects are relevant and they
are usually parametrized by introducing the crossover variable
z = aN1/2 1− T Tθ (ln N)−4/11 (1.9)
which is normalized in terms of the behaviour of the second virial coefficient Γ2. Indeed, for T → Tθ, i.e. z → 0, one sets
Γ2 = ˆ R3 G 2 (4π) 3/2z + O(z2), (1.10)
1At present the most accurate estimate of ν is 0.587597(7), N. Clisby, Phys. Rev. Lett.
where ˆRG is the zero-density radius of gyration. The z variable parametrizes
the crossover between the θ-behaviour (z ≈ 0) and the good-solvent be-haviour (z → ∞).
1.2.1
D ilut e r egim e
The first case we treat is the dilute regime. Let c be the chain number density. A system made of polymers in a good solvent is said to be in the dilute regime when the coils are mostly separated from each other, i.e. when c ˆRdG∼ cNνd≪ 1 (1.11)
where ˆRG is the zero-density radius of gyration and N is the degree of
poly-merization, i.e. the number of monomers.
Figure 1.4: Schematic representation of polymer molecules in dilute solution. The main feature is that overlaps between chains are practically absent. [31]
The osmotic pressure Π is one of the most easily accessible quantities in polymer physics. In the dilute regime we can expand it in powers of c, obtaining the so-called virial expansion:
βΠ
c = 1 + X
n=1
Γn+1cn, (1.12)
where the coefficients Γn, called virial coefficients, depend on T , N and
chem-ical details of the polymer solution. In the good-solvent regime, the ratios Γn+1/ ˆR3nG converge to universal constants as N → ∞. In [14] the first four
terms of the virial expansion of the osmotic pressure in the good-solvent regime are given, of which we report for simplicity the scaling limit (i.e. large N) form: βΠ c ≈ 1 + 1.313ηp+ 0.559η 2 p − 0.122η3p, (1.13) where β = 1/kBT and ηp ≡ 4π ˆR3 G 3 c (1.14)
is the so-called polymer packing fraction. In terms of ηp the dilute limit is
defined as ηp .1. We can equivalently write equation (1.13) in terms of the
second virial coefficient Γ2 as
βΠ
c ≈ 1 + X + 0.324X
2− 0.054X3, (1.15)
where X ≡ Γ2c.
Figure 1.5: The compressibility factor Z ≡ βΠ/c vs φp = ηp for several
values of z in the dilute region (ηp .1). The solid line labelled “GS” is the
good-solvent (z → ∞) behaviour [15].
As we approach the θ-point the virial coefficients decrease and for T → Tθ,
see Fig. 1.6. For T 6= Tθ and finite N crossover effects are important. For instance Γ2(T, N) ˆ R3 G(T, N) =− 4 11 (4π)1/2 lnN + (4π)3/2 2 z + O(ln −2 N, z2). (1.16) Similar expressions hold for all Γn+1(T, N).
Figure 1.6: Second virial coefficient ˆΓ2 (cm3/g) vs temperature (◦C) for
polystyrene of different molecular weights in a decalin solution. The molec-ular weight of the polymer chains decreases going from top to bottom. All lines intersect at the θ-temperature t ∼ 15◦C [6].
We remark that expression (1.12) is not the only one used in the literature. Experimentalists especially use another form for the virial expansion which
is basically the expansion of the compressibility in terms of the weight con-centration ρ, i.e. MΠ RT ρ = 1 + X n=1 ˆ Γn+1ρn (1.17) where ˆ Γn+1 = Γn+1 N M n , (1.18)
M is the molar mass of the polymer, R is the gas constant and N is Avo-gadro’s number.
1.2.2
Sem i-dilut e r egim e
A system made of polymers in a good solvent is said to be in the semi-dilute regime when
c ˆRdG ∼ cNνd&1, (1.19)
but still the monomer concentration cmonis approximately zero. Since cmon=
Nc ∼ N1−νdη
p ∼ N−0.8ηp, for any finite ηp, cmoncan be made arbitrarily small
by taking N large enough.
Hence we are now dealing with more concentrated solutions, where coils start to overlap (see Fig. 1.7). We may define the polymer number density c∗ which
corresponds to the onset of overlap as 4π
3 c
∗Rˆd
G = 1. (1.20)
In the good-solvent semidilute regime we can write, for N large enough,
Z ≡ βΠ
c = f (ηp). (1.21)
For ηp → ∞, Z has the asymptotic behaviour [18]
Z ∼ η1/(3ν−1)
p ∼ ηp1.311 ∼ X1.311. (1.22)
The function f (ηp) has been determined in a number of studies. Ref. [14]
Figure 1.7: Crossover between dilute and semi-dilute regimes: (a) dilute, (b) onset of overlap, (c) semi-dilute [18].
Z ≈ 1 + 1.5260η p+ 0.7954ηp2 1 + 0.5245ηp 1.311 . (1.23)
This equation is very accurate in the dilute regime because it reproduces exactly the virial expansion (1.15). Further, it agrees very well with experi-ments (see Fig. 1.8).
Besides the osmotic pressure, Caracciolo et al. [15] computed the virial ex-pansion for the radius of gyration in the semidilute regime in the scaling limit, obtaining R2 G ˆ R2 G ≈ (1 + 0.655ηp+ 0.28ηp2)−0.115. (1.24)
Figure 1.8: Comparison of the inverse compressibility K = ∂(βΠ)/∂c ob-tained by using Eq. (1.23) (“interp”) and by fitting different sets of data for polystyrene (“expt”) (see Ref. [14] and Ref. [47]).
1.3
M ixt ur es of colloids and non-adsor bing
p olym er s
Mixtures of colloids and non-adsorbing polymers show a quite interesting phase diagram, with the possibility of fluid-fluid transitions which are not present in purely colloidal systems. It is customary to distinguish two cases: the colloid limit in which the size of the polymers is smaller than that of the colloids and the protein limit in which the opposite holds.
1.3.1
Colloid lim it
In this section we report the results of the experiments of W. Poon et al. [56], who studied the behaviour of a mixture of colloids and polymers in a solvent near the θ-point. They considered size ratios q ≡ RG/Rc < 1, where RG
is the radius of gyration of polymer coils and Rc the radius of a stabilised
colloid.
The system is made of PMMA spheres dispersed in cis-decahydronaphtalene (cis-decalin) and stabilised with a layer of grafted non-adsorbing polymers
as described in Sec. 1.1.
The polymers employed to stabilise the colloids are made of chemically grafted poly-12-hydroxystearic acid (PHSA). Cis-decalin is a good solvent for PHSA at room temperature (the experiment has been done at T = 292 K), hence PHSA provides steric stabilization. The interaction strength between PHSA brushes grafted onto PMMA spheres has been measured experimen-tally by using atomic force microscopy (see Ref. [17]). This work showed that, as soon as two layers start to interpenetrate by few nanometers, the potential rises to many kBT .
Polystyrene (PS) is then added to the colloidal suspension. The θ-tempera-ture for PS in cis-decalin is TPS
θ = 286 K. The experiment is done at T = 292
K, hence very close to Tθ. The strength of the interaction between PMMA
and PS was not measured. However, it is a priori plausible that they should be mutually non-adsorbing [56].
Figure 1.9: Phase diagram of PMMA colloids and PS polymers dispersed in cis-decalin at 292 K at three size ratios q (ξ in the figure). Horizontal axis: colloid volume fraction ηc (φ in figure); vertical axis: polymer concentration
ρ (g/cm3, c
p in figure). Circles = fluid, diamonds = gas + liquid, crosses =
gas + liquid + crystal, plus signs = liquid + crystal, squares = gas + crystal, triangles = aggregation/gel, stars = glass [56].
The equilibrium phase diagrams of PMMA particles with Rc ∼ 220 nm mixed
with PS for three different values of q = 0.08, 0.24, 0.57 are shown in figures 1.9. They confirm the picture predicted in the theoretical studies [32, 41], in particular the fact that the phase diagram is strongly dependent on q.
For little q (∼ 0.08, see Fig. 1.9a), only two phases exist: a fluid one and a solid one. The phase diagram is therefore qualitatively similar to that of a purely colloidal system. Note however that, as predicted, the width of the fluid-solid coexistence region increases with respect to its parent hard-sphere system.
For each one of the three values of q of Fig. 1.9, samples with colloid volume fraction ηeff < ηHSF = 0.494 (which is the volume fraction corresponding to
hard-sphere freezing) remained in single phases and appeared homogeneous, with a fluid-like arrangement of colloidal particles. The sequence of events upon further addition of polymer depends critically on the size ratio q. Differences from the phase diagram of a purely colloidal system grow as the concentration c of polymers increases. As the polymer concentration c becomes larger and larger, the width of the fluid-solid region increases until a crystal phase is not possible anymore. In fact, crystallization was not observed at large concentrations for any of the three types of polymers. Instead, the samples exhibited metastable “gel” states [40].
Beside fluid-solid coexistence, for q > q∗ one also observes fluid-fluid coexis-tence, as predicted by Lekkerkerker’s Free Volume Theory [41]. The critical value q∗ has been estimated as q∗ ∼ 0.24 [56]. Above the critical value three
phases exist: a solid one, a liquid one and a gas one with gas-liquid and liquid-solid coexistence regions. A line of triple points also occurs. For val-ues of q just above q∗, the liquid-gas coexistence region is extremely narrow,
the transition being still unstable (see Fig. 1.9b). This region, however, grows as q increases further (see Fig. 1.9c).
The fact that q must exceed a critical value in order to observe liquid-gas coexistence can be heuristically explained as follows.
We anticipate from next chapter that an effective, attractive, two-body force arises between colloids due to the fact that around each colloid there exists a spherical region, called depletion region in which no polymers can be present. This attractive force is called depletion force. The peculiarity of the depletion force (or, equivalently, of the depletion potential) is that, at variance with the quantum-mechanical atomic potentials, which have a fixed range and strength, its range and strength can be tuned by varying, respectively, the polymer-to-colloid diameter ratio q and the concentration c of the polymers. The heuristic explanation of the presence of a liquid–solid transition is based
Figure 1.10: A heuristic explanation for the presence of a gas-liquid phase transition [56].
on the Lindemann criterion. Lindemann [42] noticed that melting in a generic crystal occurs when the average interparticle spacing increases by 10%. This criterion can be applied to a system of hard spheres in order to predict its volume fraction at melting ηM
HS. The increase of interparticle spacing of 10%
corresponds to an increase of occupied volume of ∼ 1/3. Since the volume fraction of hard spheres in the crystal phase is ηcpHS ≃ 0.74, following the Lindemann criterion we get ηM,LindHS ≃ 0.555 which is not far from the volume fraction of hard spheres at melting ηM
HS≃ 0.545. Hence Lindemann criterion
is reliable for hard-sphere systems (like colloids). In [56] it was argued that, in order to observe a liquid-gas phase transition, there must be a sufficiently long-ranged effective attractive potential among the colloids (see Fig. 1.10). Moreover, a dense liquid phase is possible only if the range of the attractive potential among the colloids is greater than the average particle spacing at melting. Since the average colloid distance in the crystal phase is equal to 2Rc, the distance at melting is 2.2Rc. Anticipating from next chapter that
the range of the effective interaction potential between colloids is 2(Rc+RG),
the argument described above says that a dense liquid phase is possible only if
2(RG+ Rc) & 2.2Rc ⇒
RG
Rc
= q & 0.1. (1.25) Hence, for q small no liquid colloid phase exists, in agreement with experi-ments.
1.3.2
P r ot ein lim it
The majority of theoretical and experimental works on colloid-polymer mix-tures has been done in the so called colloid limit, in which the polymer-to-colloid size ratio is q < 1. The opposite regime, in which q > 1, is called protein limit. Indeed, the solid transition that proteins undergo in the pres-ence of nonadsorbing large polymers, is similar to the fluid-solid transition in colloid-polymer solutions with q > 1.
Figure 1.11: “Mesh” view of a colloid-polymer system in the protein limit with the characteristic length ξ shown [23].
When q & 1 it is no longer useful to consider the system as made by colloids interacting by means of a polymer-induced effective potential. A more real-istic approach is obtained by considering the colloidal particles as dispersed in a semi-dilute polymer solution, such that the colloids diffuse through a polymeric “mesh” (Fig. 1.11).
The relevant length scale is now the polymer correlation length ξ, which is much smaller than the zero-density radius of gyration of a polymer chain.
Figure 1.12: Bolhuis et al. simulations: volume fraction representation (ηc, ηp)
[(φc, φp) in figure] of the phase diagram of a colloid-polymer mixture in the
protein limit for non-interacting polymers (left) and interacting polymers (right). Dashed lines (left) represent theoretical predictions, with stars mark-ing the critical points. Inset (right): The same binodals plotted in a reduced polymer density representation [10].
One of the first theoretical treatments of colloid-polymer mixtures in the protein limit has been done by de Gennes, who showed that the free energy for the insertion of a single hard, non-adsorbing sphere of radius Rc into an
athermal polymer solution scales as
βFc ∼ (Rc/ξ)3−1/ν, (1.26)
where ν is the Flory exponent, ξ = ˆRGηp−ν/(3ν−1) ≈ ˆRGηp−0.77 is the polymer
correlation length, and ηp is the polymer volume fraction. Using the scaling
behaviour of Fc, de Gennes [18] and Odijk [51] predicted extensive miscibility
for colloid-polymer mixtures in the large q limit if Rc < ξ. However, it is
well known that protein-polymer mixtures do phase separate. To clarify this situation, Bolhuis et al. [10] performed computer simulations.
They modelled the polymers as walks on a simple cubic lattice and consid-ered both interacting and non-interacting walks: in the first case polymers were represented as self-avoiding walks, in the second one as random walks.
In both models there was also an excluded-volume interaction among the colloids and the polymer segments.
The results of their simulation are presented in Fig. 1.12. First, the au-thors note that both models show extensive immiscibility, in agreement with experiments. Second, the two systems exhibit striking differences. In the non-interacting case the critical colloid volume fraction η∗
c tends to zero with
increasing size ratio q, while in the interacting case it is nearly constant. For both systems the critical polymer concentration η∗
p increases with increasing
q.
Van Duijneveldt et al. [23] made experiments to determine the phase dia-grams in the protein limit for q > 2. They considered small silica particles stabilized by a grafted layer of stearyl alcohol, mixed with non-adsorbing polystyrene in benzene for q = 2 [24]. A similar system, with toluene instead of benzene, was studied for q = 4.1 and 5.2 [25]. A subsequent study was made of silica nanoparticles suspended in cyclohexane for q = 4.8 [26]. The phase diagrams for q = 5.2 and T = 20◦C and T = 35◦C are reported
in Fig. 1.13. The critical point was found at ηc ≃ 0.085 and ηp ≃ 2.5. An
analogous study for the size ratio q = 4.1 finds a critical point at ηc ≃
0.13 and ηp ≃ 2. In Fig. 1.14 the results for q = 4.8 [26] are compared
to the results determined by van Duijneveldt et al. [25], Tuinier et al. [66] and simulations by Bolhuis et al. [10], showing that the phase boundary lies between the predicted boundaries for interacting and non-interacting polymers (after Bolhuis et al. simulations [10]). The reasons for these quite large differences are unknown and need further work.
Figure 1.13: Phase diagram for q = 5.2 at 20◦C and 35◦C (φ = η
c, c/c∗ = ηp).
•: sample demixing at both temperatures, ◦: stable at both temperatures, △: stable at 35◦C and two phases at 20◦C . The solid line is the estimated
binodal at 35◦C and the dashed line is the estimated binodal at 20◦C. ∗
indicates the estimated critical point at 35◦C (Letters denote samples) [23,
25].
Figure 1.14: Phase diagrams for experimental studies (q = 3.2([66]), 4.2(△[25]), 4.8(◦[26]) and Bolhuis et al. simulations [10] (q = 3.85) for interacting polymers (dashed) and noninteracting polymers (dotted). The asterisks indicate critical points, where known [23, 26].
Chapt er 2
T he A OV m odel
2.1
T he m odel
In 1954 Asakura and Oosawa [3] proposed a simple model which describes mixtures of colloids and polymers, taking into account the asymmetric in-teractions between the two species. Even though it is very simple in its assumptions, it is able to catch the physics of a colloid-polymer mixture. In particular, it reproduces the phase diagram of the system, which shows three different phases (solid, liquid and gaseous), a gas-liquid critical point and a line of triple points. In this model, proposed independently by Vrij in 1976 [75], colloids and polymers are treated as spheres of radii Rc and Rp,
respectively. Interactions between two colloids and between a colloid and a polymer are hard-sphere-like. Polymers can intersect and behave as an ideal gas. Thus, the interaction potentials are:
ucc(r) = ∞ for r < 2R0 otherwise c ucp(r) = ∞ for r < R0 otherwise c+ Rp
upp(r) = 0,
(2.1)
where r is the distance between the centers of the two particles. Equations (2.1) define what nowadays is called the AOV model. Note that the energy of an allowed AOV configuration (i.e. a set of coordinates {rc} and {rp} for
which there are no forbidden overlaps) is always zero. Hence the predicted phase separation [33, 41] for this model is not due to particle attraction but has instead an entropic origin.
Note that the minimum allowed distance between a colloid and a polymer is Rcp ≡ Rc + Rp. Thus, in the AOV model every colloid is surrounded by a
region in which there cannot be polymer centres. In three dimensions this region is a sphere superimposed over each colloid, having diameter Rcp and
called depletion region.
Let us consider now the simple situation of a box of volume V in which there are Nc colloids and any number of polymers. Let us suppose that we want to
insert a polymer in this box. If Nc = 0 we can insert the polymer wherever
we want, hence the free volume Vf available to the polymer coincides with
the volume of the box: Vf = V . If the box already contains a colloid the
free volume is smaller, Vf = V − Vδ, where Vδ = 4πR3cp/3 is the volume of
the depletion region. The physics becomes interesting when the box contains a number of colloids Nc ≥ 2. Let us consider the case in which Nc = 2.
We can distinguish two different situations. In the first case the two colloids are well separated, in such a way that their respective depletion regions do not intersect. Hence the free volume accessible to the polymer is Vf = V −
2Vδ. In the second case the colloids are close so that their depletion regions
intersect; the free volume accessible to the polymer increases: V − 2Vδ <
Vf < V . This has immediate physical consequences: by aggregating—i.e. by
staying at a distance r such that 2Rc < r < 2Rcp—the colloids can increase
the free volume Vf accessible to polymers, hence the total entropy of the
system. Under certain conditions the gain in entropy is sufficient to drive phase separation [73].
2.2
D eplet ion-induced phase separ at ion
Let us consider a colloid dispersion. If we add non-adsorbing polymers to it we observe a new phenomenon: an attractive force arises between the col-loids, which increases as the concentration of the added polymers increases. Understanding the mechanism which underlies this attractive force is an in-teresting challenge, because it is this force that drives the phase separation of the mixture.
Depletion-induced phase separation is a convenient way of concentrating col-loidal dispersions. It can also be used for large-scale size fractionation in industrial applications. Beside these practical reasons, colloid-polymer mix-tures are interesting systems in which there is an easy method of changing
the range and depth of the interaction potential between the colloidal parti-cles, for instance by varying the polymer-to-colloid size ratio or the polymer density.
When the attraction is strong enough (i.e., above a certain polymer concen-tration) a phase transition occurs, which is very similar to the one we observe in atomic systems, for instance in Argon.
Figure 2.1: Osmotic pressure around a colloid. For a single sphere this pressure is isotropic (a). However, if two depletion layers overlap, the osmotic pressure becomes anisotropic and there is a net attractive force, as indicated by the arrows (b) [65].
The mechanism underlying the depletion interaction can be explained by con-sidering a pair of spherical colloids in a solution of non-adsorbing polymers, as it is depicted in Fig. 2.1. The average equilibrium polymer concentration decreases when going from the bulk (the maximum segment concentration) towards the sphere surface (where the concentration is zero due to the pres-ence of the depletion layer). In a simplified treatment one can replace the polymer concentration profile with a Heaviside step function. The zero part of the step function corresponds to a layer in which the polymer concentra-tion is zero (the depleconcentra-tion layer), as indicated by the (dashed) layers around the spheres in Fig. 2.1. Outside this layer the polymer concentration is equal to the average bulk polymer concentration. Under this approximation it makes sense to consider polymers as hard spheres (as long as polymer-colloid
interactions are concerned) as the AOV model does.
The discovery of the depletion interaction is due to Asakura and Oosawa in 1954 [3]. They showed that two plates immersed in a solution of ideal non-adsorbing polymers attract each another with a force which arises because of excluded-volume reasons. This was actually the first theory of depletion interactions. Their work was extended by Vrij in 1976 [75]. The main ideas are the following.
Figure 2.2: Effective attraction between two hard spheres of diameter σ2 due
to the presence of hard spheres of diameter σ3 [75]
Let us consider the effective interaction between a pair of spheres of diameter σ2 in a dilute suspension of spheres of diameter σ3 ≤ σ2. The interaction
potential of two hard spheres (i, j) is given by
uij(r) = ∞ for r ≤ ¯σ0 for r > ¯σ , σ¯ ≡ (σi+ σj)/2. (2.2)
If n3 is the concentration of the particles of diameter σ3, for n3 → 0 the
potential of the mean force v(r) between the large particles is equal to u22(r).
How will v(r) change when n3 is finite?
It is evident from Fig. 2.2 that no small sphere can lie in the intersection of two depletion regions, giving rise to a pressure unbalance that in turn is responsible of the depletion interaction. Let us compute this unbalance, following the procedure of Vrij [75].
We assume that the spheres of diameter σ3 are dilute so that their osmotic
A B O C r/2 σ
Figure 2.3: Computation of the unbalance of osmotic pressure.
σ2 + σ3, the pressure does not act on the whole colloid; in particular there
is no pressure on the dashed part of the surface (see Fig. 2.3). Therefore, there is a force f equal to P0S, where S is the area of the circle going
through points A and B and orthogonal to OC. It is straightforward to obtain S = π(¯σ2 − (r/2)2) so that f =−πn3kBT ¯σ2 1− r 2 4¯σ2 . (2.3)
We have introduced a minus sign to take into account the fact that the force is attractive. By integrating we obtain the depletion potential
v(ρ) =−v0 1− 3 4ρ + 1 16ρ 3 , (2.4) where ρ = r ¯ σ , v0 = 4 3π¯σ 3n
3kBT , valid for ρmin =
σ2
¯
limit (strictly only for q . 0.15) the colloid-colloid attractive interaction is pairwise additive. This is not the case in the protein limit, as many body interactions are now important due to multiple interactions between a single polymer coil and more than a colloidal particle.
2.3
Fr ee Volum e T heor y
The approximate analysis of the AOV model was performed by Lekkerkerker et al. in 1992. This approach is called Free Volume T heory (FVT). It allows us to derive the phase diagram analytically and it is able to reproduce the whole phase diagram of a mixture of colloids and polymers. In particular, it predicts the presence of the solid, liquid and gaseous phases.
Let us work in the grand canonical ensemble. If we fix the colloid positions, the polymer contribution is simply given by
Ξ(~rc,1,· · · , ~rc,n) = X p eµpNp Np! Z(N p, ~rc,1,· · · , ~rc,n), (2.6)
where Z is the canonical partition function of a system of Np polymers,
which are ideal gas particles that move in a free volume Vfree({~rc,n}). The
grand-partition function is immediately computed giving
Ξ(~rc,1,· · · , ~rc,n) = expβΠp(µp)Vfree({~rc,n}), (2.7)
where Πp(µp) is the pressure of an ideal gas of particles at chemical potential
µp.
Hence if we integrate out the polymer degrees of freedom we end up with an effective interaction between the colloids given by
W = Uc− Πp(µp)Vfree(~rc), (2.8)
where Uc is the interaction potential between colloids. The free volume
de-pends clearly on the positions of the colloids {~rc}, in particular it depends
on their number Nc and on the degree of the overlap of their depletion zones.
Up to now, no approximations are made. To go further, one applies the so-called Van der Waals mean-field approximation. The idea is to substitute the free volume Vfree accessible to the polymers, which is a function of the
coordinates of every colloid, with its average value in the “unperturbed” system constituted by the only colloids. This means rewriting
Vfree = α(ηc, q)V, (2.9)
where α(ηc) represents the average fraction of the total volume of the box in
which polymers can move. It depends on the colloid volume fraction
ηc = 4 3πR 3 c Nc V (2.10)
and on the ratio q ≡ Rp/Rc = (Rcp− Rc)/Rc.
Equivalently we can work in the canonical ensemble and use N and V as control parameters. In this case the relevant thermodynamic potential is Helmholtz’s free energy. Reasoning as above, we obtain for a system of Nc
colloids and Np polymers in a volume V
F = Fc(Nc, V ) + Fp(Np, α(ηc)V ). (2.11)
The free energy decouples in a term which corresponds to a system made only by colloids in a volume V and in a second term corresponding to a system made only of polymers in a volume α(ηc)V . The interaction between colloids
and polymers is only due to the function α(ηc).
Polymers are treated as an ideal gas, so that their free energy is:
Fp(Np, α(ηc)V, T ) =−kBT Npln eVfree Npλ3 = =−kBT Npln e λ3 − kBT Npln α(ηc)V Np . (2.12) We can neglect the first term because the terms which are linear in Np and
V do not contribute to the derivation of the phase diagram. If we define np ≡ Np/V , we obtain [41] Fp(Np, α(ηc)V, T ) = kBT npV ln np α(ηc) + other terms. (2.13)
In order to determine the expression of Fc(Nc, V ) we use the precise available
Z(T, V, N) = p(T, V, N)V NkBT
. (2.14)
Since pressure and Helmholtz’s free energy are related by ∂F (T, V, N) ∂V T,N =−p(T, V, N), (2.15) integrating between V0 and V , we obtain
− Z V
V0
p(T, ξ, N) dξ = F (T, V, N)− F (T, V0, N)≡ ∆F. (2.16)
Since Z is an intensive quantity, for a colloidal system it only depends on ηc.
Hence ∆F = NckBT ηc Z ηc,0 Z(ηc′)dη ′ c η′ c . (2.17)
To compute the phase diagram we have to specify a particular form for the compressibility Z(ηc) depending on the phase we are considering. For the
fluid phase we use Carnahan and Starling’s approximation for the compress-ibility of a hard-sphere liquid [16],
Zliquid(ηc) =
1 + ηc+ η2c − ηc3
(1− ηc)3
, (2.18)
while for the solid phase Hall’s expression [35] might be used:
Zsolid(ηc) = (12− 3ψ)/ψ + 2.557696 + 0.1253077ψ
+ 0.1762393ψ2− 1.053308ψ3+ 2.818621ψ4
− 2.921934ψ5+ 1.118413ψ6, (2.19)
where ψ = 4(1− ηc/ηccp), ηccp ≃ 0.74 is the close-packing volume fraction for
hard spheres.
Finally we need the excluded volume fraction α(ηc, q). A phenomenological
α(ηc, q) = (1− ηc) exp(−Aγ − Bγ2− Cγ3), (2.20) where γ = ηc 1− ηc , A = 3q + 3q2+ q3, B = 9 2q 2 + 3q3, C = 3q3. (2.21)
Let us calculate now the polymer and colloid chemical potentials. Let us start from µp(T, V, N) which can be computed as
µ(T, V, N)≡ ∂F (T, V, N) ∂N T,V . (2.22)
In our case we have to differentiate only the polymeric part of the free energy because Np is absent in the colloidal part:
µp(Np, α(ηc)V, T ) = ∂Fp(Np, α(ηc)V, T ) ∂Np T,V . (2.23) We find µp kBT = ln np α(ηc) . (2.24)
For the colloid chemical potential, instead, we have to differentiate also the polymeric part of the free energy because of the presence of the term α(ηc)
which contains implicitly Nc. We obtain
µc kBT = Z Z ηc dηc+ Z− 4 3πR 3 c np α dα dηc . (2.25)
For the total osmotic pressure Π = −∂F
∂V we find 4πR3 cΠ 3kBT = ηcZ + 4 3πR 3 c np α α− ηc dα dηc . (2.26)
At coexistence µp must be identical in both phases. If we define nr = np/α,
we obtain nr = exp µp kBT ≡ zp, (2.27)
where zp is the fugacity of the polymers. Hence, nr must be continuous
along the coexistence line. Then we express µc and Π as a function of ηc and
nr. Then, to compute the two colloid concentrations η(l)c and ηc(g) along the
liquid-gas coexistence curve, we solve the system of equations (
µc(η(l)c , nr) = µc(ηc(g), nr)
Π(ηc(l), nr) = Π(η(g)c , nr)
. (2.28)
Explicitly, the first equation can be written as Z ηc(g) ηc,0 Z(ξ) ξ dξ+Z(η (g) c )− 4 3πR 3 cnrα′(ηc(g)) = Z η(l)c ηc,0 Z(ξ) ξ dξ+Z(η (l) c )− 4 3πR 3 cnrα′(ηc(l)), (2.29) that is Z η(l)c η(g)c Z(ξ) ξ dξ + Z(η (l) c )− Z(ηc(g)) + 4 3πR 3 cnr h α′(ηc(g))− α′(ηc(l))i = 0. (2.30)
Analogously, the second condition becomes
ηc(l)Z(ηc(l))−η(g) c Z(ηc(g))+ 4 3πR 3 cnr h α(η(l)c )−α(η(g) c )+ηc(g)α′(ηc(g))−ηc(l)α′(ηc(l)) i = 0. (2.31) If we solve equations (2.30) and (2.31) we obtain the colloid packing fractions ηc(l) and ηc(g) as a function of nr. From these we derive the corresponding
polymeric volume fractions as η(i)p = 4πR3pα(η(i)c )nr/3.
The corresponding phase diagrams are shown in Figs. 2.5 and 2.6. They qualitatively agree with the numerical results for the AOV model, see Ref. [8].
Figure 2.5: Phase diagram of AOV model in the (ηc, ηpr) representation for
q < 0.3 (left) and for q > 0.3 (right) [41].
Figure 2.6: Phase diagram of AOV model in the (ηc, ηp) representation for
Chapt er 3
Simulat ion M et hod
3.1
G r and Canonical M ont e Car lo
We perform extensive Monte Carlo simulations of a colloid-polymer system in the grand-canonical ensemble. In this ensemble the control parameters are the volume V , the chemical potentials µc, µp (temperature does not play any
role since the system is athermal), while the number of particles of the two species (respectively, Nc for the colloids and Np for the polymers) fluctuate.
The equilibrium probability distribution for the number of particles is given by the usual grand-canonical distribution:
P (Nc, Np) = 1 Ξz Nc c zpNpe−βH, (3.1) where Ξ(V, zp, zc) = P Np,Nc zNp p zNccQ(V, Np, Nc), Q(V, Np, Nc) is the
configu-rational partition function of a system of Np polymers and Nc colloids in
the volume V and zi = eβµi are the fugacities. We normalize the partition
functions as Q(V, 1, 0) = Q(V, 0, 1) = V so that zp and zc are
dimension-ful constants. In the following we report them in units of Rc, hence zcR3c
and zpR3c. In terms of the standard fugacity definition they correspond to
zc(Rc/λ)3 and zp(Rc/λ)3.
The aim of a grand-canonical Monte Carlo simulation is to generate config-urations of the system according to the probability distribution (3.1). The most common update move used in grand-canonical Monte Carlo simulations consists in the insertion and removal of a single particle, according to the fol-lowing algorithm (we consider only the hard-sphere case for which e−βH is
either 0 or 1; the generalization is trivial): I nser t ion:
• we randomly choose a point ~r ∈ V ;
• we propose the insertion of the particle in the point ~r;
• we check the hard-sphere constraints and reject the move if they are not satisfied;
g(τ → σ) = 1 Nc+ 1
. (3.4)
The detailed balance condition can be written as
P(σ)g(σ → τ)A(σ → τ) = P(τ)g(τ → σ)A(τ → σ). (3.5) A solution is provided by Metropolis et al. [48] and is given by
A(σ → τ) = min 1,P (τ ) P (σ) g(τ → σ) g(σ → τ) . (3.6)
Substituting the previous expressions we obtain
Aci = min 1, zcV Nc + 1 , Api = min 1, zpV Np + 1 , Acr = min 1, Nc zcV , Apr = min 1, Np zpV . (3.7)
This algorithm is not particularly efficient when the colloid volume fraction is high, because it becomes difficult to insert a polymer or, even worse, a colloid inside the box. Hence it is necessary to develop a move that is more efficient, i.e. that has an acceptance probability ≃ 10 ÷ 30% also for high-density configurations.
Efficient MC moves were developed by Vink [73]. They are called in the literature cluster moves because they involve groups of particles rather than single ones. Here we describe them:
Colloid inser t ion and p olym er r em oval This move is made of the following steps: • we randomly choose a point ~r ∈ V ; • we propose the insertion of a colloid in ~r;
• we check the hard-sphere constraints and reject the move if they are not satisfied;
• we remove the polymers that are within the depletion zone of the new colloid;
• we accept it with probability Ai to be computed below.
Figure 3.1: Vink cluster insertion move: in the configuration (A) a colloid is inserted (B) and all the polymers which lie in its depletion region are removed (C) [73].
The difference with the previous case is that here, beside inserting a colloid, all polymers that are within the depletion zone of the new proposed colloid, i.e. inside the sphere centered in ~r and having radius Rcp, are removed (see
Fig. 3.1). We must then define an inverse move, i.e. a move which consists in removing a colloid and, consequently, inserting polymers. Let us suppose we want to remove a colloid. The idea is to insert a certain number of polymers in the depletion zone centered in the point in which the colloid was situated. But how many polymers?
The idea of Vink consists in choosing the number np of polymers to be
inserted uniformly in the interval [0, mmax). Note that this is a finite interval,
hence at most mmax−1 polymers are inserted. This is not consistent with the
colloid insertion algorithm we have introduced before since in that case any number of polymer can be removed. Hence, to guarantee detailed balance, before checking the hard-sphere constraints, we perform the following check: • we count the polymers which occur in the depletion zone of the new colloid; if their number np is such that np ≥ mmax the move is rejected.
In principle any value mmax can be used: the algorithm satisfies detailed
balance in all cases. However, in order to have an efficient MC dynamics it is important that mmax be larger than the typical number of polymers present
in the depletion zone. Since polymers are ideal-gas particles we expect their density to be of order zp. Hence the number of polymers in the depletion
volume Vδ = 4πR3cp/3 is zpVδ with fluctuations of order pzpVδ. Hence, Vink
proposes mmax = 1 + max h 1, intzpVδ+ αpzpVδ i . (3.8)
Here α is a positive constant of the order of unity: α ∼ 2 generally gives good results [73]. Now it is possible to introduce the removal move (see Fig. 3.2):
Figure 3.2: Vink cluster removal move: a colloid (A) is removed and r poly-mers (B) are randomly put in its depletion region (C) [73].
Colloid r em oval and p olym er inser t ion The move is made of these steps:
• we randomly choose a colloid;
• we generate a random integer number r ∈ [0, mmax);
• we randomly choose r points ~ri ∈ Vδ, where Vδ is the depletion region
of the chosen colloid;
• we place r polymers in these points, checking each time the hard-sphere constraints (if they are not satisfied the move is rejected);
• we accept the move with probability Ar to be computed below.
It is easy to figure out that this algorithm is more CPU-time consuming than the previous one. Nevertheless, it has a much better efficiency, because the dynamics is reasonably fast even for high colloid volume fractions.
Let us compute now the acceptance probabilities Ai and Ar. As in the
previ-ous case we first consider the move in which a colloid is inserted. Let us call σ the initial state and τ the final state. The state σ is characterized by Nc
colloids and Np polymers. The state τ , instead, is characterized by Nc + 1
colloids and Np− np polymers, where np is the number of polymers present
within the depletion sphere centred on the colloid. The grand-canonical prob-abilities are (again we assume that τ satisfies the hard-sphere constraints):
P (σ) = 1 Ξz Nc c zNpp, P (τ ) = 1 Ξz Nc+1 c zpNp−np. (3.9)
Let us compute now the proposal probabilities. When we insert a colloid in the box, the only freedom we have is in the choice of the insertion point, hence
g(σ → τ) = 1
V . (3.10)
The expression for the probability of the inverse transition (τ → σ) is a little bit more complicated. First, we choose the colloid, which brings a factor
1
Nc+1; then we must choose the number np of polymers to be inserted, which
brings a factor 1/mmax. Then we choose np points within the volume of the
depletion zone which gives a factor 1/Vnp
δ ; finally note that a permutation
of these np points corresponds to the same move which gives an additional
factor np!. Finally we get:
g(τ → σ) = np!
mmax(Nc + 1)Vδnp
. (3.11)
Using the Metropolis prescription (3.6), the probability Ai for the colloid
insertion move is Ai = min 1,P (τ ) P (σ) g(τ → σ) g(σ→ τ) = min 1, zcV np! mmax(Nc+ 1)(zpVδ)np . (3.12)
Ar = min 1,mmaxNc zcV (zpVδ)np np! . (3.13)
3.2
Simulat ed Tem p er ing
The classical grand-canonical Monte Carlo method described above is not appropriate for the study of systems undergoing first-order phase transitions: to overcome these difficulties we have used the umbrella sampling method combined with the simulated tempering algorithm [44]. They are described below.
3.2.1
Simulat ed t em p er ing
In the simulated tempering method one considers a wider configuration space. Each configuration is specified by an integer m ∈ [1, Nm], where Nm is a
parameter, and by the positions {~ri} of the polymers and the colloids.
In the usual grand canonical ensemble each configuration has probability 1
Ξz
Np
p zcNce−βH({~r}), (3.14)
where Ξ is the partition function Ξ(V, zp, zc) =
X
Np,Nc
zNp
p zcNcQ(V, Np, Nc). (3.15)
In the extended ensemble, to a configuration (m,{~r}) we associate the prob-ability 1 ΞSTfmz Np p zNc,mc e−βH({~r}), (3.16) where ΞST =X m fmΞ(V, zp, zc,m). (3.17)
Note that we use a different colloid fugacity for each m, while the polymer fugacity is m independent. Hence, in order to completely specify the system, we should choose Nm and a set of Nm colloid fugacities. It is important to
note that grand-canonical averages for any zc,m can be obtained from averages
in the extended ensemble. For any observable O we have
hOiGC,zc,a =hOδmaiST (3.18)
where zc,a is one of the chosen colloid fugacities and h·iGC is the usual grand
canonical average.
The constants fm should be chosen so that all terms in the sum are
approx-imately equal. If we require
fmΞ(V, zp, zc,m) = fm−1Ξ(V, zp, zc,m−1), (3.19) we obtain fm fm−1 = Rm, fm−1 fm = Sm, (3.20) with Rm ≡ * zc,m−1 zc,m Nc+ m , (3.21) Sm ≡ * zc,m zc,m−1 Nc+ m−1 , (3.22)
where h·im indicates the mean value with respect to the partition function Ξ
with zc = zc,m. Combining these expression we define the ratios as
fm
fm−1
=pRm/Sm. (3.23)
The simulated tempering method, as implemented here, works well close to the critical point, but is inefficient in the two-phase region. For this reason, we have combined the method with the umbrella sampling technique which is discussed in the next section.
3.3
U mbr ella sam pling
In the presence of first-order transitions (as it is the case with the AOV model) it is quite difficult to sample correctly the Grand Canonical (GC)
distribution (3.1). To bypass the difficulties we consider the umbrella sam-pling method [64].
The main idea of this method is to simulate a different equilibrium distribu-tion with respect to the GC one, i.e.
1 π(Nc)
zNp
p zNcce−βE, (3.24)
where π(Nc) is an appropriate function which will be defined below. If h·iGC
and h·iπ are the averages with respect to the GC distribution and to the
dis-tribution (3.24), respectively, we have for an arbitrary observable O(Nc, Np):
hO(Nc, Np)iGC = hπ(Nc
)O(Nc, Np)iπ
hπ(Nc)iπ
. (3.25)
This relation allows us to obtain GC averages from simulations using the dis-tribution (3.24). The function π(Nc) (called in the literature “the umbrella”)
must be chosen so that in the simulation the system can move easily between the two phases. Consider the histogram of Nc in the GC distribution, i.e.
h(Nc,0) =hδ(Nc, Nc,0)iGC, (3.26)
where δ(x, y) is Kronecker’s delta. Assume that the system is close to phase separation so that h(Nc) has two peaks at Nc,min (colloid-gas phase) and
Nc,max (colloid-liquid phase). The optimal choice is then
π(N) = ah(Nc,min) N ≤ Nc,min,
π(N) = ah(N) Nc,min ≤ N ≤ Nc,max,
π(N) = ah(Nc,max) N ≥ Nc,max,
(3.27)
where a is an irrelevant constant. Indeed, if Nc,min ≤ Nc ≤ Nc,max, the
observed histogram in the umbrella distribution is flat, i.e. independent of Nc. Hence, the system can move freely between the two phases, allowing a
precise determination of any required thermodynamic property.
As in the simulated tempering scheme described above, we consider Nm
col-loid fugacities {zc,m}, such that for zc,1(zc,Nm) the system is in the colloid-gas
(colloid-liquid) phase. Then, we determine the umbrella functions πm(Nc)
it-eratively. First, we perform a short hysteresis cycle in which we perform Ntherm GC iterations at zc = zc,1, then at zc,2, and so on, up to zc,Nm; then we
decrease zc till we reach again zc,1. If h(1),+m (Nc) and h(1),−m (Nc) are the
his-tograms obtained at z = zm (the + refers to the distribution obtained while
increasing zc and the − to that obtained while decreasing the fugacity), we
set h(1)m (Nc) = h(1),+m (Nc) + h(1),−m (Nc) and πm(1)(Nc) = h(1)m (Nc)/M if h(1)m (Nc)≥ M, πm(1)(Nc) = 1 if h(1)m (Nc)≤ M (3.28) where M ≡ maxNc[h (1)
m (Nc)]/10. Then, we repeat again the same hysteresis
cycle several times. At iteration k, for each zc,m we perform the simulation
using the distribution (3.24) with π = πm(k−1). Then, we set
π(k)m (Nc) = πm(k−1)(Nc)h(k)m (Nc)/M if h(k)m (Nc)≥ M,
π(k)m (Nc) = πm(k−1)(Nc) if h(k)m (Nc)≤ M,
(3.29)
where M ≡ maxNc[h (k)
m (Nc)]/10. We stop when we observe that, for at least
some values of m, h(k),+m (Nc) and h(k),−m (Nc) are non-vanishing in an interval
of values of Nc that extends between the two phases.
Once we have a reasonable estimate of the functions πm(Nc), we could just
perform an extensive simulation at a single value of zc,m (an optimal choice
would be to take the value for which πm(Nc) is clearly bimodal). Data for
dif-ferent values of zc could just be obtained by standard reweighting techniques.
However we have found more convenient to use all information we have collected and simulate all systems together, using the simulated-tempering method. Note that, in the standard implementation of the method, one should be careful that the fugacities zc,m are such that the colloid-number
distributions overlap; otherwise, no fugacity swap is accepted. In our case, since we use umbrella distributions, the overlap condition is always verified, and thus the number Nm of needed systems is always small. Typically we
take Nm = 10. If Ξπm(V, zp, zc,m) = X Np,Nc zNp p zc,mNc πm(Nc) Ξ(V, Np, Nc), (3.30)
we consider the extended partition function ΞST =X
m
The constants fm are again chosen so that all terms in the sum are
approxi-mately equal. If we require
fmΞπm(V, zp, zc,m) = fm−1Ξπm−1(V, zp, zc,m−1), (3.32) we obtain fm fm−1 = Rm, fm−1 fm = Sm, (3.33) with Rm ≡ * zc,m−1 zc,m Nc πm(Nc) πm−1(Nc) + π,m , (3.34) Sm ≡ * zc,m zc,m−1 Nc πm−1(Nc) πm(Nc) + π,m−1 , (3.35)
where h·iπ,m indicates the mean value with respect to the umbrella
distribu-tion (3.24) with zc = zc,m, π = πm . Combining these expressions we define
the ratios as
fm
fm−1
=pRm/Sm. (3.36)
The constants Rm and Sm are determined together with the umbrella
sam-pling functions πm. Then, we set f1 = 1 and use Eq. (3.36) to determine the
constants fm, m≥ 2.
In the presence of disorder—we will specify the model in detail in the next Chapters—the GC partition function is still given by Eq. (3.24), with the only difference that one should take into account the interactions between the freely moving particles and the matrix. Since the GC partition function depends on the matrix, also the functions πm and the constants fmare matrix
dependent. Thus, we recompute them when we restart the simulation with the different matrix.
In the simulation we determine the colloid and polymer histograms for a large number (typically 100) of colloid fugacities zc,r. They are obtained by
mea-suring, for each matrix realization α, the reweighted histograms pc(α, zc,r, Nc,0)
pc(α, zc,r, zc,m, Nc,0) = X i zc,r zc,m Nc,i
πm(Nc,i)δ(Nc,i, Nc,0)δ(zc,m, zc,i),
pp(α, zc,r, zc,m, Np,0) = X i zc,r zc,m Nc,i
πm(Nc,i)δ(Np,i, Np,0)δ(zc,m, zc,i),(3.37)
where i refers to the MC iteration, and Np,i, Nc,i, zc,i are the number of
polymers and colloids and the colloid fugacity at the ith iteration. The colloid histogram is then
hc,ave(Nc,0, zp, zc,r) = 1 Nα X α pc(α, zc,r, zc,m, Nc,0) P Ncpc(α, zc,r, zc,m, Nc) , (3.38)
where Nα is the number of matrix realizations. Note that we obtain a
differ-ent estimate of the distributions at zc,r for each of the zc,m.
Finally, note that our estimates (3.38) are biased, since they are disorder averages of a ratio of thermal averages. This means that, if we take the limit Nα → ∞ at fixed Niter, we obtain estimates that differ from the correct result
by a term (the bias) of order 1/Niter. One could perform a bias correction, as
discussed in Ref. [36]. However, given the small number of disorder instances we consider, we have found that in the present case the bias correction is not relevant.
Chapt er 4
T he A OV m odel wit h disor der
4.1
I nt r oduct ion
The study of the properties of a fluid in a porous material is of theoretical and practical interest. These materials are characterized by a highly inter-connected porous structure. They have important technological applications, for instance in catalysis and gas separation and purification1. Examples are
the Vycor glasses, in which pore sizes range from 1 nm to 100 nm, and high-porosity systems like silica gels (xerogels and aerogels), which are produced by means of silica sol-gel processes.
The existence of a network of walls and pores introduces an interaction be-tween fluid and walls (which represent the “impurities”). The competition between this interaction and the fluid-fluid interaction can lead to interesting surface-driven phase changes [34]. The physical challenge consists in under-standing how the presence of the porous structure modifies the properties of the system with respect to the case without impurities (also known as “bulk”). For instance, when the dimensions of the pores are comparable to the range of the intermolecular forces, the average number of coordina-tion of the molecules falls dramatically, and we expect to see large shifts in the coexistence curves, which is experimentally seen (see [34] and refer-ences therein). Furthermore, the presence of pores changes the topology of the system and sometimes the space dimensionality. For a slit-shaped pore 1For a list of experimental studies of binary mixtures in porous materials, see the
references cited in E. Sch¨oll-Paschinger, D. Levesque, J.-J. Weis, and G. Kahl, Phys. Rev. E 64, 011502 (2001).
structure, for instance, the reduction of the pore width changes the system from three-dimensional to two-dimensional.
The confinement of a fluid in a porous matrix has practical applications in many fields of science such as geology, agriculture, ceramics, biomedicine, petroleum engineering and powder metallurgy [61]. For instance, almost fifty-percent of the original oil-in-plane is left in a typical oil reservoir by conventional extraction techniques. Recovering this oil is an appealing issue for industry. Porous materials are a modern research field in Materials Sci-ence too: many brand-new materials like aerogels have a porous structure, by virtue of which they gain very particular properties: in 2002 a new version of aerogel, employed on NASA’s Stardust spacecraft, has been recognized by Guinness World Records as the solid with the lowest density [49]. Another practical problem tied to the study of fluids in porous media is the restoring of contaminated aquifers [61]. Also wood is a porous material.
In this chapter we study the demixing of colloid-polymer mixtures in porous materials, with the purpose of understanding how the binodals and the crit-ical properties of the mixture change in the presence of impurities. AOV colloid-polymer mixtures in a porous matrix have been studied in Refs. [55, 63, 68, 69, 74] by means of density-functional theory, integral equations, and Monte Carlo (MC) simulations. The nature of the critical transition has been fully clarified [55, 68, 69, 74]: if obstacles are random and there is a preferred affinity of the quenched obstacles to one of the phases, the transition is in the same universality class as that occurring in the random-field Ising model, in agreement with a general argument by de Gennes [19]. If these conditions are not satisfied, standard Ising or randomly dilute Ising behaviour is ob-served instead, see Refs. [20, 28]. On the other hand, little is known on how demixing is influenced by the amount of disorder and by its nature (for a polymer matrix some results for the critical-point behaviour as a function of the amount of disorder are reported in Ref. [68]). This is the main purpose of the present chapter.
4.2
D efi nit ions
We remind from Chap. 2 that in the AOV model [3, 4, 73, 75] colloids and polymers are treated as spheres of radii Rc and Rp, respectively. Interactions
hard-sphere-like. Polymers can intersect and behave as an ideal gas. Thus, the interaction potentials are: ucc(r) = ∞ for r < 2Rc 0 otherwise ucp(r) = ∞ for r < Rc+ Rp 0 otherwise upp(r) = 0, (4.1)
where r is the distance between the centers of the two particles.
Disorder has been introduced by considering a colloidal quenched matrix which has a hard-sphere interaction both with the colloids and the polymers. In practice, we choose a disorder concentration cdis and randomly distribute
Ndis = cdisL3 nonoverlapping spheres of radius Rdis in a cubic box of size L.
The position of these spheres is assumed to be fixed (quenched). Colloids and polymers can only move outside the quenched matrix, which means that the spheres belonging to the matrix and the freely moving particles interact with pair potentials
uc,dis(r) = ( ∞ for r < Rc + Rdis, 0 for r≥ Rc+ Rdis, up,dis(r) = ( ∞ for r < Rp + Rdis, 0 for r≥ Rp+ Rdis. (4.2)
Note that the matrices considered here are different from those discussed in Refs. [68, 69]. The main difference is that here the matrix consists in hard spheres that cannot intersect each other (we name it colloidal matrix). On the other hand, in Refs. [68, 69] the matrix spheres are soft and can freely overlap, as if they were an ideal gas (hence the name polymer matrix). Second, in those works, for a given choice of cdis, the number Ndis was not
fixed, but obtained from a Poissonian distribution with mean value cdisV ,
where V was the volume of the system. This second difference should not be important in the infinite-volume limit, since it entails density fluctuations of order 1/√V , which vanish as V → ∞.
4.3
T he eff ect ive volum e fr act ion
f
In the simple model we consider, disorder is characterized by two parameters, the reduced concentration ˆc≡ cdisRc3—or, equivalently, the volume fraction
ηdis—and the ratio Rdis/Rc. However, ˆc does not directly characterize the
free space available to colloids and polymers. We use instead the effective matrix filled-space ratio f , which is defined as follows. Consider the region R in which the (centers of the) colloids are allowed:
R = {r : |r − ri| ≥ Rc+ Rdis, for all 1≤ i ≤ Ndis }, (4.3)
where ri is the position of the i-th hard sphere belonging to the matrix. If
VR is the volume of the region R, we define
f ≡ 1 −[VR]
L3 , (4.4)
where [VR] is the average of VR over the different matrix realizations. Note
that, for large values of L, [VR] is essentially independent of the matrix
real-ization, a property known as self-averaging. The parameter f represents the volume fraction that is unavailable to the colloids due to the presence of the random matrix and can easily be determined by computing the probability of inserting a colloid in the otherwise empty matrix. In a completely analogous way we can define fpol, which characterizes the volume fraction unavailable
to polymers. Of course, f > fpol in the colloid regime in which q < 1, while
f < fpol in the opposite, protein regime.
It is interesting to understand qualitatively how the disorder distribution changes with Rdis at fixed f . In Fig. 4.1 we show the matrix for f = 0.5
and two values of Rdis, Rdis/Rc = 0.1 and Rdis/Rc = 3. To make the figure
more clear, we consider a two-dimensional system, that is a matrix of non-overlapping disks on a square of area L2. It is evident that the topology of the
matrix is quite different. For large Rdis/Rc the free volume available to the
colloids consists in large empty regions connected by narrow channels. This is the case of a porous material with big interconnected pores. On the other hand, for Rdis/Rc small, pores are significantly smaller and the topology of
the network is more complex (see also Fig. 4.2 and Fig. 4.3).
In order to have demixing, the parameter f cannot be arbitrarily close to 1, but should satisfy f < f∗, where f∗ is related to the percolation threshold
Figure 4.1: Two-dimensional systems on a square L2 with L/R
c = 50 and
f = 0.5. On the left we take Rdis/Rc = 0.1, while on the right Rdis/Rc = 3.0.
The disorder packing fractions πcR2
dis are 0.0057 and 0.292 in the two cases,
respectively. The red (gray) circles of radius Rc + Rdis correspond to the
colloid-excluded region (depletion region) around each sphere of the quenched matrix: the centers of the colloids can only belong to the white region. We also draw a single colloid (black) to show the length scale.
Table 4.1: Estimates of the reduced concentration ˆc ≡ cdisR3c and of the
disorder packing fraction ηdis ≡ 4πR3discdis/3 for two values of f and several
values of Rdis/Rc.
f = 0.40 f = 0.70 Rdis/Rc cˆ ηdis ˆc ηdis
0.005 0.120 6.3· 10−8 0.283 1.5· 10−7 0.01 0.118 4.9· 10−7 0.279 1.2· 10−6 0.02 0.115 3.9· 10−6 0.271 9.1· 10−6 0.05 0.105 5.5· 10−5 0.248 1.3· 10−4 0.1 0.0915 3.8· 10−4 0.215 9.0· 10−4 0.2 0.0700 2.3· 10−3 0.164 5.5· 10−3 0.4 0.0431 0.0116 0.0972 0.026 0.6 0.0280 0.0254 0.0609 0.055 1.0 0.0136 0.057 0.0278 0.116
in disconnected finite regions and thus no phase transition is possible. The exact value of f∗ is unknown. However, the arguments of Ref. [62], which we
will present in detail in Sect. 4.4, suggest
f∗ ≈ 0.85. (4.5)
For the same reasons—polymers should be able to move in the whole space— the polymer parameter fpol must satisfy fpol < f∗ in order to observe
coex-istence.
We shall perform simulations for two values of f , f = 0.40 and f = 0.70, the latter being quite close to the threshold f∗, and for q = 0.8, so that
fpol < f . In Table 4.1 we report the reduced concentration ˆc and the disorder
volume fraction ηdis ≡ 4πRdis3 c/3 for several values of Rdis/Rc. A plot of
f as a function of Rdis/Rc for several ηdis is reported in Fig. 4.4, while a
contour plot is given in Fig. 4.5. First, we observe that ˆc converges to a finite positive constant as Rdis/Rc → 0. This result is quite easy to understand.
If Rdis/Rc ≪ 1, the pair potentials (4.2) become essentially independent of
Rdis. Hence, the density becomes essentially independent of Rdis for Rdis
small. In the opposite limit Rdis/Rc ≫ 1, the potentials become essentially
Figure 4.2: A realization of the matrix of impurities for (ηdis, Rdis/Rc) =
(0.01, 0.3).
fraction ηdis. For instance, for ηdis = 0.30, we obtain f = 0.51, 0.40, 0.32 for
Rdis/Rc = 5, 10, 50. Since a liquid hard-sphere phase exists only up [38] to
η ≈ 0.49, for large Rdis/Rc, the matrix may belong to different hard-sphere
phases, while still satisfying the condition f < f∗.
Finally, as appears from Fig. 4.4, the function f (ηdis, Rdis/Rc) at fixed ηdis is
very well fitted by the functional expression
f (ηdis, Rdis/Rc) = 1− exp
" −a(ηdis) Rc Rdis − b(ηdis) Rc Rdis 3# , (4.6)
![Figure 1.10: A heuristic explanation for the presence of a gas-liquid phase transition [56].](https://thumb-eu.123doks.com/thumbv2/123dokorg/4780887.48324/19.892.286.636.179.503/figure-heuristic-explanation-presence-gas-liquid-phase-transition.webp)
![Figure 1.14: Phase diagrams for experimental studies (q = 3.2([66]), 4.2( △[25]), 4.8(◦[26]) and Bolhuis et al](https://thumb-eu.123doks.com/thumbv2/123dokorg/4780887.48324/23.892.279.633.632.900/figure-phase-diagrams-experimental-studies-bolhuis-et-al.webp)
