# Homogenization of Mutually Immiscible Polymers Using Nanoscale Effects: A Theoretical Study

Authors: Sarah Montes; Agustín Etxeberria; Javier Rodriguez; Jose A. Pomposo
Journal: Research Letters in Physical Chemistry
 (2008)

---

## Abstract

A theoretical study to investigate homogenization of mutually immiscible polymers using nanoscale effects has been performed. Specifically, the miscibility behavior of all-polymer nanocomposites composed of linear-polystyrene (PS) chains and individual cross-linked poly(methyl methacrylate)-nanoparticles (PMMA-NPs) has been predicted. By using a mean field theory accounting for combinatorial interaction energy and nanoparticle-driven effects, phase diagrams were constructed as a function of PMMA-NP size, PS molecular weight, and temperature. Interestingly, complete miscibility (i.e., homogeneity) was predicted from room temperature to 675 K for PMMA-nanoparticles with radius less than ~7 nm blended with PS chains (molecular weight 150 kDa, nanoparticle volume fraction 20%) in spite of the well-known immiscibility between PS and PMMA. Several nanoscale effects affecting miscibility in PMMA-NP/PS nanocomposites involving small PMMA-nanoparticles are discussed.


---

## Body

## 1. Introduction

Nanoscale effects are responsible
for the growing scientific interest in both nanomaterials and nanocomposites [1]. Often, nanoobjects display interesting physical
phenomena such as surface plasmon resonance in metallic nanoparticles (NPs), ballistic
transport in carbon nanotubes (CNTs), or fluorescence emission in quantum dots
(QDs), among others. Most of them are useful for a huge of potential applications
covering from nanoelectronic to improved sensors devices [2]. Also, nanocomposites composed of well-dispersed
nanoobjects into a polymer matrix lead to unexpected mechanical [3] and/or rheological properties [4]. A representative example is the large decrease (up
to 80%) in melt viscosity upon nanoparticle addition first observed in polystyrene-nanoparticle
(PS-NP)/linear-polystyrene (PS) blends by Mackay et al. [5]. The underlying physics of this unexpected behavior
is being currently explored [6] and should be rationalized
in terms of scaling concepts at the nanoscale [7].
Blend miscibility (homogeneity) is a necessary condition to observe such a viscosity
drop in all-polymer nanocomposites (i.e.,
polymer-nanoparticle/linear-polymer blends).Recently, we have introduced
an entropic model for predicting the miscibility behavior of PS-NP/PS
nanocomposites with very good agreement between theory and experiment [8]. Additionally, the theory has been employed for
the prediction of the interaction parameter, the miscibility behavior, and the
melting point depression of athermal poly(ethylene) (PE)-nanoparticle/linear-PE nanocomposites using chain dimensions data from Monte-Carlo (MC)
simulations [9]. Our main findings indicate
that dilution of contact, hard sphere-like, nanoparticle-nanoparticle
interactions plays a key role in explaining the miscibility behavior of polymer-nanoparticles
dispersed in a chemically identical linear-polymer matrix [8, 9].In a recent work, the athermal
model has been extended to calculate the phase diagram of weakly interacting
all-polymer nanocomposites by accounting for combinatorial interaction energy
and nanoparticle-driven effects [10]. Complete
miscibility was predicted for PS-nanoparticles with radius < 6 nm blended
with poly(vinyl methyl ether) (PVME) at low concentrations. When compared to linear-PS/PVME blends displaying phase splitting at T>375 K, the miscibility
improving effect of sub-10 nm PS-nanoparticles was clearly highlighted [10].In this letter, we
explore theoretically the conditions for homogenization of two mutually
immiscible polymers by changing all the linear-polymer chains of one of the
components by cross-linked polymer nanoparticles. Specifically, we consider the
PS/poly(methyl methacrylate) (PMMA) pair as a model system. Immiscibility
between PS and PMMA is well known in the literature as a result of unfavorable
interactions between styrene (S) and methyl methacrylate (MMA) repeat units
[11–13]. Here, miscibility diagrams for PMMA-NP/PS nanocomposites are reported
as a function of PMMA-NP size, PS molecular weight (Mn) and
temperature. Finally, several nanoscale effects affecting the miscibility
behavior of PMMA-NP/PS nanocomposites are also discussed.


## 2. Theoretical Approach

For a binary blend to be
thermodynamically stable against phase separation, the following well-known conditions
must be fulfilled:(1)Δgmix<0,Δgmix(2)≡(∂2Δgmix∂ϕ12)>0. In a mean field, theoretical framework, Δgmix, the free energy of mixing (per
unit volume) for an all-polymer nanocomposite composed of spherical
polymer-nanoparticles (component 1) of volume fraction ϕ1, radius Rp, nanoparticle volume υ1, and linear-polymer chains of
degree of polymerization N2(≫1) and monomer volume υ2 is given by [10]
1Δgmix=Δgmixco+Δgmixnp+Δgmixex,1aΔgmixco=kT[ϕ1υ1ln⁡ϕ1+ϕ2N2υ2ln⁡ϕ2],1bΔgmixnp=kT[ϕ1υ1(4ϕ1−3ϕ12(1−ϕ1)2)+32×ϕ1ϕ2N2υ2(RpRg2)2],1cΔgmixex=(r2Rp)[kTυ2(χS+χHT)]ϕ1ϕ2,
where k is the Boltzmann constant, T is the absolute temperature, Rg2 is the radius of gyration of
linear-polymer 2, r2 is the radius of a repeat unit 2,  and χS and χH are the entropic and enthalpic components of
the blend interaction parameter (χ=χS+χH/T), respectively.Equation (1a) provides
the contribution to the total free energy of mixing due to the combinatorial entropy of mixing, whereas (1b)
gives the contribution to the free energy of mixing due to nanoparticle-driven
effects. Equation (1b) takes into account the dilution of hard sphere-like nanoparticle-nanoparticle interactions upon mixing (first RHS term) [8, 9] and the stretching of the linear-polymer chains due to the presence of the nanoparticles (second RHS term). Referring to this latter term, a Ginzburg-type expansion term is adopted to account for the fact that the
polymer-nanoparticles cause stretching of the polymer chains in their vicinity [14].Equation (1c) accounts
for the temperature-dependent
and nanoparticle size-dependent interaction effects in the all-polymer nanocomposite. A prefactor (r2/Rp)
is introduced in (1c) since the number of surface contacts with monomers 2 for
each nanoparticle becomes smaller as one increases the nanoparticle radius [14]. This ratio tends toward unity inasmuch Rp→r1 as it should, r1 being the radius of a repeat unit 1.For the sake of simplicity, in (1) we have omitted a term (~constant×ϕ1) arising from the (non-zero) reference free energy of the pure component 1 in the disordered molten state at   ϕ1fr=0.494 (maximum packaging of monodisperse spheres) as dicussed in [9] which has no effect in the resulting spinodal equation (second derivative of the free energy of mixing with respect to  ϕ1).The
conditionΔgmix(2)=0 just determines the spinodal
miscibility boundary in the phase diagram, which is given by 2Δgmix(2)=Δgmixco(2)+Δgmixnp(2)+Δgmixex(2)=0,2aΔgmixco(2)=kT[1υ1ϕ1+1N2υ2ϕ2],2bΔgmixnp(2)=kT[2υ1(4−ϕ1(1−ϕ1)4)−92υ2(Rpr2N2)2],2cΔgmixex(2)=−2(r2Rp)[kTυ2(χS+χHT)], where we have employed Rg2≈2r2N21/2/6.Equations (1)–(2) are
presumably valid only for ϕ1<0.494 (the theoretical maximum
nanoparticle packaging volume fraction without freezing) and for Rp>~5r2 (it should be noted that r2 is typically around 0.3 nm) [10].


## 3. Results and Discussion

It is well known that PS and PMMA
lead to heterogeneous (phase separated) blends due to the immiscibility between
components at high molecular weights [11–13]. The S/MMA temperature-dependent
interaction parameter (χ=χS+χH/T) has been determined by several experimental techniques such as small
angle neutron scattering (SANS) (using deuterated block copolymers) [11, 12] and
cloud-point (CP) measurements (using oligomer mixtures and taking into account
end-group effects) [13] with good agreement between them. As an example, χ=0.028+3.9/T by SANS measurements [11] and  χ=0.021+2.8/T by
CP experiments [13]. Since the
entropic contribution, χS, is much greater than the (χH/T) term, the temperature dependence of χ was found to be relatively weak.We have employed the
temperature-dependent interaction parameter determined by CP measurements to
calculate the spinodal miscibility boundary of PMMA-NP/PS composites as a
function of nanoparticle radius at constant blend composition (see Figure1). Complete
miscibility across the 275–675 K temperature range was predicted for
PMMA-nanoparticles with radius less than 6.8 nm in spite of the well-known
immiscibility between PMMA and PS homopolymers. Conversely, for PMMA-nanoparticles
with radius higher than 7.2 nm, complete immiscibility (phase separation) is
expected. For PMMA-NP of radius in between 6.8 and 7.2 nm, partial miscibility
was predicted as a function of temperature (the blends displaying upper
critical solution temperature (UCST)-type behavior). No significant changes was
observed when χ values from SANS experiments [11, 12] were
employed in the calculations.Figure 1
Predicted phase diagram for PMMA-NP/PS (Mn=150 kDa) nanocomposites as a function of
nanoparticle size as calculated from (2) by using  χ=0.021+2.8/T [13], ϕ1=0.2, N2=1443,υ2=99 cm3/mol, and r2=0.32 nm.In order to rationalize
these nanoscale-driven results, we have examined the values of the different
factors governing (2) (second derivative of the energy of mixing). Δgmixco(2) (arising from combinatorial effects) was found to be positive (favorable to
mixing) and increased linearly with temperature. Conversely, Δgmixex(2) (arising from interactions) was
negative and increased in absolute value with T. In general, |Δgmixex(2)|>Δgmixco(2) so miscibility was conditioned by favorable values of the Δgmixnp(2) term. At constant ϕ1, the size of the PMMA-nanoparticles
was the main factor affecting the Δgmixnp(2) term (see (2b)). At a given temperature, Δgmixnp(2) was found to decrease upon increasing the PMMA-NP size leading, respectively, to
partial and complete immiscibility at PMMA-NP radius of 7 and 7.5 nm. As a
result, miscibility in PMMA-NP/PS nanocomposites can be mainly attributed to two
combined effects: (1) reduced (unfavorable) PS chain stretching by smaller PMMA-nanoparticles
and (2) favorable dilution of (hard sphere-like) nanoparticle-nanoparticle
interactions upon mixing.The effect of PS
molecular weight on the predicted phase diagram for PMMA-NP/PS nanocomposites
is illustrated in Figure2. As expected, a reduction in PS molecular weight
shifts the miscibility boundary toward smaller nanoparticle sizes due to the
large entropy penalty paid for nanoparticle inclusion into short linear-polymer
chains. Hence, the critical PMMA-NP radius for complete immiscibility (RpC) changes from 5.4 to 7.2 and 7.8 nm upon changing the PS molecular
weight from 50 to 150 and 500 kDa, respectively.Figure 2
Influence of PS molecular weight on the calculated phase
diagram for PMMA-NP/PS nanocomposites (ϕ1=0.2): Mn=50 kDa (solid
squares), Mn=150 kDa (solid circles), and Mn=500 kDa (solid triangles).Concerning the effect of
blend composition onRpC for PMMA-NP/PS (Mn=150 kDa) nanocomposites, a linear
increase in RpC (up to ~29%) on going from ϕ1=0.15 (RpC=6.9 nm) to ϕ1=0.35 (RpC=8.9 nm) was observed.


## 4. Conclusions

A mean field theoretical model,
accounting for combinatorial interaction energy and nanoparticle-driven
effects, has been employed to investigate homogenization of mutually immiscible
polymers using nanoscale effects. The PS/PMMA pair has been selected as a
model system since immiscibility between PS and PMMA is well documented in the
literature, and reliable values of the S/MMA interaction parameter are
available as a function ofT. Specifically, we have investigated the effect on blend
miscibility (homogeneity) of the replacement of linear-PMMA chains by
cross-linked individual PMMA-nanoparticles.Hence, phase diagrams have
been constructed for PMMA-NP/PS nanocomposites as a function of PMMA-NP size,
PS molecular weight, blend composition, and temperature. Interestingly, complete
miscibility across the 275–675 K temperature range was predicted for
PMMA-nanoparticles with radius less than 6.8 nm blended with PS (Mn=150 kDa, ϕ1=0.2). Increasing PS molecular weight and
nanoparticle content was found to have a small positive effect on PMMA-NP/PS
nanocomposite miscibility.Homogenization of PMMA-NP/PS nanocomposites was mainly attributed to two combined nanoeffects: reduced PS chain
stretching by the smaller PMMA-nanoparticles, and favorable dilution of contact
(hard sphere-like) nanoparticle-nanoparticle interactions upon mixing.



---

*Source: 504917-2008-06-09.xml*