Physica A 388 (2009) 3093–3099

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Physica A

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Demixing of amphiphile–polymer mixtures investigated usingdensity functional theoryLing Zhou a,∗, Bing-hua Sun a, Zhong-ying Jiang b, Jian-guo Zhang a, Hai-ying Gu aa School of Science, Nantong University, Nantong 226007, Chinab College of Physics and Electronic Information, Yili Normal Institute, Yining 835000, China

a r t i c l e i n f o

Article history:Received 6 January 2009Received in revised form 10 March 2009Available online 10 April 2009

Keywords:DemixingAmphiphilePolymerDensity functional theory

a b s t r a c t

We construct a functional for amphiphile–polymer mixtures and investigate the demixingtransition by using a proposed version of density functional theory. It is found that increaseof the amphiphilic size ratio and polymer length can effectively promote phase separationof the systems. Phase diagrams are plotted to clarify these influences. The results providean effective way of controlling the stability of the fluid–fluid phase equilibrium of themixtures.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Amphiphile and polymer mixtures exist widely in Nature. In recent years, much attention has been paid to the self-assembly of such systems because of their numerous applications in biology and industry. In many experiments, intensivephase separation and rich phase behavior have been observed. Examples are provided by DNA and surfactant complexes[1], hydrophobically modified or charged polymer and surfactant mixtures [2] and so on. In the course of controllingmicrostructures, how to stabilize the system is a fundamental problem. However, only limited understanding has beenobtained. So far, relatively little theoretical work has been done on this problem. Moskalenko et al. [3] have used asimplemean-field theory based on the Blume–Emergy–Griffiths three-spin-state latticemodel to describe three-componentmixtures of water, polymer and amphiphile, where the influence of amphiphile concentration has been considered. Tofurther address the influence of amphiphilic length, amphiphilic strength and polymer configuration, more theoretical workis needed. In the present work, we propose another model for constructing a functional for amphiphile–polymer mixtures.The functional is constructed by combining the Schmidt density functional theory (DFT) for amphiphiles [4] with the Yu andWu DFT for polymeric fluids [5]. The solvent molecule is implicit and taken as a continuous medium. Each amphiphile ismodelled as a hard sphere (head) with an infinitely thin needle (tail) which is attached radially to the sphere. Visually,the shape of the amphiphile is very like a lollipop. Though it is a simple model and only hard core pair interaction isconsidered, we note that it is sufficient for generating mesoscopic structures [6]. So the advantage of this model is thatwe can investigate the basic phase behavior solely driven by entropy, which effectively eliminates the influence of otherinteractions. In addition, considering many technological applications, we find that it is of interest to predict and improvethe stability of a givenmixture. Obviously, for an amphiphile–polymermixture, it may undergo a demixing phase transitioninto amphiphile-rich and polymer-rich phases under certain conditions. As an application of our proposed functional, weinvestigate the phase separation of amphiphile–polymer mixtures and reveal the trends in phase stability determined by

∗ Corresponding author.E-mail address: ZL7103@163.com (L. Zhou).

0378-4371/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2009.04.006

3094 L. Zhou et al. / Physica A 388 (2009) 3093–3099

Fig. 1. Sketch of the binary mixture of amphiphiles and polymers.

amphiphilic strength and polymer length. The results are accordance with experiments [7]. Without loss of generality, oursystem reduces to a rod–polymer mixture [8] or a sphere–polymer mixture [9,10] if the length of amphiphile tails tends toinfinity or zero compared with the radius of the heads.

2. Model and method

We consider a mixture of amphiphiles and homopolymers. As mentioned above, we use hard body model to mimicamphiphiles. The head diameter is σ and the tail length is L. The centre of the head (denoted by r) is taken as the positionof the amphiphile and the direction of the tail is given by a unit vector�; see Fig. 1. The polymers are represented by freelyjointed hard sphere chains composed of M tangentially connected segments of the same diameter σ . Sphere–sphere andsphere–needle overlaps are forbidden. The excluded volume between two needles is zero and the needle–needle interactionis assumed to be ideal. However, we should point out that the possibility of an isotropic–nematic transition cannot bepredicted due to this simplification of the amphiphile tail with a vanishing volume.In the framework of density functional theory, the total configurational free energy of the system is a functional of the

local densities of polymers, ρM(R), and amphiphiles, ρA(r,�):

F [ρM(R), ρA(r,�)] = Fid[ρM(R), ρA(r,�)] + Fex[ρM(R), ρA(r,�)], (1)

where R ≡ (r1, r2, . . . , rM) stands for a set of coordinates of polymers reflecting the segmental positions.The ideal Helmholtz free energy Fid is given by

βFid[ρM(R), ρA(r,�)] = β∫dRρM(R)Vb(R)+

∫dRρM(R)[ln(ρM(R))− 1]

+

∫dr∫d2�4π

ρA(r,�)[ln(ρA(r,�))− 1], (2)

where β = (kBT )−1, kB is Boltzmann’s constant, T is temperature. Vb(R) is the total bonding potential and satisfiesexp[−βVb(R)] =

∏M−1i=1 δ(|ri+1 − ri| − σ)/4πσ 2.

The excess Helmholtz free energy Fex is obtained by the integration of the excess free energy density:

Fex =∫dr∫d2�4π[Φhs + Φp + ΦAp]. (3)

In the above, Φhs, Φp and ΦAp represent the excess free energy densities reflecting the packing effects of hard spheres, theconnectivity of polymer chains and the hard core repulsion between hard spheres and needles, respectively.

L. Zhou et al. / Physica A 388 (2009) 3093–3099 3095

The hard sphere partΦhs is constructed using the Fundamental Measure Theory (FMT) of Rosenfeld [11,12], given by

Φhs = −n0ln(1− n3)+n1n2 − nv1 · nv2

1− n3+ (n32 − 3n2nv2 · nv2)

n3 + (1− n3)2ln(1− n3)36πn23(1− n3)2

, (4)

where the nα (α = 0, 1, 2, 3, v1, v2) denote weighted densities evaluated using spatial convolutions nα(r) =∫dr′ρp(r′)wα(r− r′)+

∫dr′∫ d2�′

4π ρA(r′,�′)wα(r− r′). ρp(r) is the polymer average segment density defined as ρp(r) =∑M

j=1

∫dRδ(r− rj)ρ(R). The weight functionswα(r) are connected to the geometrical properties of spheres via

w3(r) = Θ(σ/2− |r|), w2(r) = δ(σ/2− |r|),w1(r) = w2(r)/(2πσ), w0(r) = w2(r)/(πσ 2),wv2(r) = δ(σ/2− |r|)r/|r|, wv1(r) = wv2(r)/(2πσ). (5)

Θ(r) is the Heaviside step function, and δ(r) is the Dirac delta function.Besides the influence of the direct bonding potential included in Eq. (2) for polymers, there exists a correlation between

nonbonded polymeric segments due to chain connectivity. It arises from the excluded volume effects between segmentsand is considered in the second term on the right-hand side of Eq. (3). Wertheim’s first-order perturbation theory [13–17]takes into account the structure of the hard sphere reference system and provides a relatively accurate description ofthermodynamic properties. In this work, we use the version proposed by Yu and Wu, who extend Wertheim’s first-orderperturbation theory for a bulk fluid to an inhomogeneous system by using FMT-style weighted densities [5]:

Φp =1−MM

n0ξ ln[yhs(σ , nα)], (6)

where ξ = 1−nv2 ·nv2/n22 is an inhomogeneous factor, and yhs(σ , nα) is the contact value of the radial distribution function

(RDF) for the hard sphere fluid given by

yhs(σ , nα) =1

1− n3+

n2σξ4(1− n3)2

+n22σ

2ξ

72(1− n3)3. (7)

To specify the sphere–needle contribution,ΦAp is given by Schmidt’s DFT [18] for the hard rod–sphere mixtures:

ΦAp = −n̄0 ln(1− n3)+n̄1n̄21− n3

, (8)

where the needle local weighted densities are

n̄ν(r,�) =∫dr′ρA(r′,�)w̄ν(r− r′,�), ν = 0, 1, (9)

and the ‘‘mixed’’ sphere–needle weighted density is

n̄2(r,�) =∫dr′∫d2�′

4πρA(r′,�′)w̄2(r− r′,�)+

∫dr′ρp(r′)w̄2(r− r′,�). (10)

The corresponding weighted functions for the needle part are

w̄0(r,�) =12[δ(r+ (R+ L)�)− δ(r+ R�)], (11)

w̄1(r,�) =14

∫ R+L

Rdlδ(r+ l�), (12)

and the weighted function for the sphere–needle part is

w̄2(r,�) = 2|wv2(r) · �|, (13)

where R = σ/2 is the radius of the spheres.These complete the description of the total free energy.

3096 L. Zhou et al. / Physica A 388 (2009) 3093–3099

Fig. 2. Phase diagram in the ηA–ηp plane with q = 10, 15, 20. The polymer length is fixed at M = 10. The binodals, spinodals and critical points aredenoted by solid lines, dot–dashed lines and black dots, respectively.

Fig. 3. The coexistence lines for the systems from Fig. 2 in the mole fraction (x)–inverse pressure (1/p∗) plane.

3. Results

To study the demixing behavior of the mixtures, we consider the stability of the bulk phase. In this case, the vectorweighted densities vanish and the scalar weighted densities are proportional to the bulk densities ρp and ρA. The total freeenergy per unit volumeΦv can be written asΦv = Φhs +Φp +ΦAp + β−1ρA(ln ρA − 1)+ β−1ρp(ln ρp − 1). FromΦv , thepressure p and the chemical potentials of the two species can be calculated using

βp = −Φv +∑j=A,p

ρj∂Φv

∂ρj, βµj =

∂Φv

∂ρj. (14)

In our calculations, we vary two parameters: the size ratio q and the polymer length M , where q is defined as the ratiobetween the tail length L and the head radius R of the amphiphiles. The amphiphilic strength is controlled by themagnitudeof q. As q increases, the amphiphilic strength increases. It is found that the system can demix entropically into amphiphile-rich (polymer-poor) and polymer-rich (amphiphile-poor) phases under appropriate conditions. To elucidate the full phase

L. Zhou et al. / Physica A 388 (2009) 3093–3099 3097

Fig. 4. Phase diagram in the ηA–ηp plane with M = 10, 30, 100. The size ratio is fixed at q = 10. The binodals, spinodals and critical points are denotedby solid lines, dot–dashed lines and black dots, respectively.

diagrams of the system, we would require calculating the phase equilibrium between amphiphile-rich and amphiphile-poor phases. The coexisting equilibrium densities (binodals) are obtained by solving simultaneously equations for the equalpressures and chemical potentials of the two phases, i.e., pI = pII and µ

(i)I = µ

(i)II for i = A, p. The spinodals are determined

by the condition det[∂2Φv/∂ρi∂ρj] = 0, i, j = A, p. The critical points located on the spinodals and the binodals are evaluatedusing the relations [19]

s3∂3Φv

∂[ρp]3+ 3s2

∂3Φv

∂[ρp]2∂ρA+ 3s

∂3Φv

∂ρp∂[ρA]2+∂3Φv

∂[ρA]3= 0, (15)

where

s ≡ −∂2Φv

∂ρp∂ρA

/∂2Φv

∂[ρp]2. (16)

First we examine the influence of the size ratio q. In Fig. 2, we plot the bulk phase diagram in the ηA–ηp plane forq = 10, 15, 20, where ηi (= πσ 3ρi/6, i = p, A) are packing fractions of polymers and amphiphiles. The polymer lengthis fixed at M = 10. The binodals, spinodals and critical points are denoted by solid lines, dot–dashed lines and black dots,respectively. The critical points are characterized by the following values of ηA and ηp: ηA = 0.100854, ηp = 0.0891229for the system with q = 10, ηA = 0.0471002, ηp = 0.0642997 for the system with q = 15 and ηA = 0.0281542,ηp = 0.0514791 for the system with q = 20. As the size ratio q increases, the curves shift left and the critical point movestowards lower amphiphile and polymer packing fractions. The results indicate that with the increase of the amphiphilic taillength, the repulsive interaction between the tail and the sphere is increased, which leads to the lessening of the miscibilityof the amphiphiles and polymers. We also show another representation of the phase diagram for the same system. Fig. 3shows the binodals in the mole fraction (x)–inverse pressure (1/p∗) plane, where x = Mρp/(ρA + Mρp) and the reducedpressure p∗ = βpσ 3. The tie lines connecting the coexisting fluid states are horizontal. For larger values of q, the observedphase boundary becomes more asymmetric and the pressure shows a dramatic change. In this case, we find that the role ofthe inverse pressure is similar to that of the temperature in the case of simple fluids. Increase of q leads to a reduction of thepressure. Therefore the demixing region is enlarged.Thenwe consider the influence of polymer lengthM on the phase equilibriumcurves. Fig. 4 shows the bulk phase diagram

in the ηA–ηp plane forM = 10, 30, 100. The size ratio is fixed at q = 10. Further increase in the polymer lengthM enlargesthe phase demixing region. The critical point also moves towards lower amphiphile and polymer packing fractions. Thecritical points are ηA = 0.100854, ηp = 0.0891229 for the system withM = 10, ηA = 0.0722897, ηp = 0.0470203 for thesystem withM = 30 and ηA = 0.0572317, ηp = 0.0245373 for the system withM = 100. Comparing with Fig. 2, we findthat increase of q more easily promotes phase demixing of the mixtures than that of M . This is attributable to the rigidityof the amphiphilic tail. Fig. 5 illustrates the critical packing fraction ηc = (ρA + Mρp) as a function of the polymer lengthM . As the polymer lengthM increases, the critical packing fraction decreases. When the polymer length is relatively short,the decrease of the critical packing fraction is very rapid. However, the opposite is found when the length of polymers issufficiently long. We find that the critical packing fraction ηc tends to a finite value in the long chain limit. This result is very

3098 L. Zhou et al. / Physica A 388 (2009) 3093–3099

Fig. 5. The critical packing fraction ηc as a function of the polymer length M . The curves from top to bottom are for q = 10 (solid line), 15 (dashed line),20 (dotted line), respectively.

similar to that for athermal colloid–polymer mixtures [20] and rod–polymer mixtures [8]. As mentioned in Section 1, oursystem reduces to the rod–polymer or sphere–polymer case when the size ratio q tends to infinity or zero. Letting q tendto infinity or zero in our system, we find that the critical packing fraction ηc converges to a finite value, which agrees wellwith the previous results [8,20].

4. Discussion

In an experimental work, Wormuth [7] has investigated the phase behavior of amphiphile–polymer mixtures inwater as a function of temperature and composition. He found that the miscibility of ethoxylated alcohol amphiphile[CH3(CH2)i−1O(CH2CH2O)jH] and poly(ethylene oxide) in water decreases dramatically with increasing amphiphilicstrength and molecular weight of the polymer. Considering the role of inverse pressure to be analogous to that oftemperature, we compare the phase diagram in Fig. 3 with the experimental one [7] in the temperature and amphiphile-plus-polymer concentration plane. The phase boundaries are qualitatively consistent with the experiments when thetemperature is lower than 40 ◦C. The chosen parameters, q andM , in ourmodel are also comparable with the strength of theamphiphiles and the molecular weight of the polymers. The promoted phase demixing due to the increase of q andM is inaccordancewith the experimental results. To obtain quantitative consistency,more sophisticatedwork is needed.We noticethatWertheim’s first-order perturbation theory gives an inaccurate description for the dilute and semidilute regimes of longchains. A direct consequence is that the calculated critical packing fraction has an error in the long chain limit. In addition,we notice that there exists a depletion interaction between spheres induced by needles when the density of amphiphiles isrelatively high. So the contact value of the RDF in an amphiphile–polymer mixture is slightly underestimated by using thecontact value of the RDF in the hard sphere fluid. However, the precise corrections are rather difficult and we leave theseproblems for future work.

5. Conclusion

In summary, we have constructed a functional for amphiphile–polymer mixtures and investigated the demixingtransition by using the proposed density functional theory. The shifts in the binodals and spinodals have been calculatedas a function of amphiphilic size ratio and polymer length. It is found that increase of the size ratio and polymer lengthcan effectively promote the phase separation of such systems. As the inverse pressure decreases, the fluid–fluid coexistencelines will broaden. When the polymer chains are very long, the limiting value of the critical packing fraction will convergeto a constant. All kinds of phase diagrams are constructed to clarify these influences. The results are consistent with theexperiments, which provide an effective way to control the stability of the fluid–fluid phase equilibrium of the mixtures.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants 10647004, 20675042, 20875051),the Natural Science Foundation of Nantong University (No. 08Z006), and the Startup Foundation for Introduced Talents ofNantong University.

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