An Equation of State for the IsotropicâNematic Phase Transition of

ARTICLE pubs.acs.org/IECR

An Equation of State for the IsotropicNematic Phase Transition of Semiflexible Polymers Yuan-Xiang Zheng, Yang-Xin Yu,* and Ying-Feng Li Department of Chemical Engineering, Tsinghua University, Beijing 100084, China

bS Supporting Information ABSTRACT: Based on the thermodynamic perturbation theory for polymer, a new equation is proposed by incorporating a wide range of molecular flexibility and perturbative interactions to describe the isotropicanisotropic (nematic) phase transition phenomena of the semiflexible polymers. In the new equation, the framework of the Helmholtz free energy of the system is the same as the Onsager-like theory. The entropy loss due to the orientation is estimated by the KhokhlovSemenov (KS) theory. With regard to the configurational free energy, the polymer is envisioned as a series of subchains. The ParsonsLee approximation is used to account for the higher virial coefficients of the subchain and Yu equation for the hard-sphere-chain fluid is adopted to modify the defect of the first order approximation in dealing with the associating points. An analytical expression of the perturbative term is obtained by employing the “square peg in a round hole” potential function and the mean-field approximation. The hard-core part of the equation reduces to the DupreYang theory when the stiffness of the molecule is high. When the model approaches the limit of the random coil, a modified equation of the hard-sphere-chain fluid is obtained. The present theory has been used to predict the isotropicnematic phase equilibrium for real semiflexible polymers with two adjustable parameters. The agreements between the theoretical results and experimental data are much better than that of previous theories.

’ INTRODUCTION Owing to its special thermodynamic, structural, and optical properties, liquid crystalline materials have wide applications in recent years, especially in the field of display. The discovery of the liquid crystal (LC) can be traced back to almost one and a half centuries ago, but the LC phase transition was not treated theoretically at a molecular level until 1950s. During that period, three totally different but all influential theories of handling the phase transition between isotropic and nematic phases were introduced for the simplest and the most frequently encountered liquid crystal structure. They are lattice theory, Onsager theory, and MaierSaupe theory. Flory1,2 used a lattice model to describe the isotropicnematic (IN) phase transition, which is very popular and has many advantages despite the introduction of an “artificial” lattice. A lot of works have been done following his idea. For example, the FloryHuggins (FH) theory,3,4 which is famous in dealing with isotropic macromolecular solutions, was extended to the IN phase transition calculation when the mixing heat cannot be neglected. The FloryRonca (FR) theory5,6 revised the twodimensional lattice model which might introduce some order even in the isotropic phase with a three-dimensional lattice. Such theories though had been studied intensively, but now are becoming less popular since they artificially neglect the nature of the molecules. A little earlier than the Flory theory, Onsager employed the so-called “virial expansion” to demonstrate the IN phase transition due to steric interactions alone7 based on the hard-rod model. Compared with the Flory theory, the Onsager theory has a solid foundation of statistical mechanics except some assumptions are introduced. Among these assumptions, the most important one is that only the second virial coefficient is retained. It should be pointed out that the Onsager theory has an analytical solution only for infinitely long rods. Attempts have been made to extend r 2011 American Chemical Society

Onsager’s seminal work to a more wide range of applications. For example, different ways to merge the higher virial coefficients were studied, including thermodynamic decoupling approximation,8,9 scaled particle theory (SPT),10 rigorous calculation through the virial expansion,11 etc. In addition, Khokhlov and Semonov12,13 applied the Lifshitz theory to formulate the orientational free energy so that the Onsager-like theory can be directly applied in the prediction for worm-like particles. The third widely accepted method to solve the IN phase transition problem is the MaierSaupe (MS) theory.14,15 This theory is totally contrary to the steric theories since it attributes the rise of the anisotropic phase to the anisotropic attractive interaction. The MS theory and its extensions are very popular for their simplicities and good characterization of the experimental data especially for thermotropic mesogens. There are extensions of this theory that incorporate excluded volume. Doi16 was first to recognize that the mathematical formulation of both Onsager and MaierSaupe theories are very similar and hence instead of working with the Onsager model, one uses the MaierSaupe model. Both types of models are approximations and rarely achieve the semiquantitative agreement with the experimental data. A lot of works have been done to incorporate the advantages of the Onsager-like and MS theories, because the theory merely embodied repulsive interaction and cannot resolve the problem thoroughly. Different models have been introduced to include the attractive interaction to the Onsager-like theory using mean field method, perturbation theory, etc.1721 It is worth mentioning here Received: November 26, 2010 Accepted: April 6, 2011 Revised: March 16, 2011 Published: April 06, 2011 6460

dx.doi.org/10.1021/ie102379h | Ind. Eng. Chem. Res. 2011, 50, 6460–6469

Industrial & Engineering Chemistry Research

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Figure 1. The schematic of the semiflexible chain used in the configurational free energy calculation. The hard core is depicted with the solid line and the upper limit of the perturbative interaction is highlighted by the dash line. The area between the two margins is characterized by the potential function given by eq 4.

that there is a completely different model, that is, Landau-de Gennes phenomenological theory,22 that is also a very popular model for dealing with NI transitions. In the last 20 years, the thermodynamic perturbation theory2327 (TPT) as well as the statistical associated fluid theory2832 (SAFT) has been developed successfully. These theories have a relatively strict theoretical foundation for freely jointed chain fluids and good descriptions of properties for the simple fluids. The hardsphere-chain model is of great interest in the IN phase transition studies.3337 The model itself can represent some important features of the hard-rod model such as the excluded volume effect. It can easily incorporate the flexibility of the polymers. For example, Williamson et al.34,35 have used the linear tangent hard-sphere-chain (LTHSC) model to study the IN phase transition, and the results showed promise when compared with Monte Carlo (MC) simulations. Hino and Prausnitz36 used the Chapman equation28 and Hu equation38 to describe the IN phase transition together with the VegaLago approximation.39 The flexibility and nonsteric interactions are important in characterizing the IN phase transition. In this work, we employ a method based on the TPT and the incorporation of the effect of the molecular flexibility and the perturbative interactions between the polymers. The semiflexible polymer is considered as a series of subchains. The ParsonsLee approximation is used to account for the higher virial coefficients of the subchain, and Yu equation40 is adopted to deal with the association. The proposed equation has the advantage of including dispersive contribution. The established equation is used to calculate the IN phase transition, and the results are compared with the experimental data.

’ THEORY The Onsager theory can be regarded as one of the earliest applications of the classical density functional theories4144 of fluids. By assuming the density distribution of the inhomogeneous fluid as F(r,Ω) = Ff(Ω), the Helmholtz free energy of the system in Onsager theory is given by F F id F res ¼ þ ¼ ½lnðνFÞ 1 þ f σ NkB T NkB T NkB T ZZ F Vexc ðΩ1 , Ω2 Þ f ðΩ1 Þ f ðΩ2 Þ dΩ1 dΩ2 þ 2

ð1Þ

where, N is the number of the molecules, F is the molecular number density of the system, kB is the Boltzmann constant, and T is the absolute temperature. The superscripts id and res represent the ideal and residual parts of Helmohltz free energy, respectively. The term [ln(νF) 1] represents the translational and rotational contribution to the free energy, that is, ν = νtνr, νt = Λ3, Λ is the de Broglie wavelength and νr = sΠni (h2/ 2πIikT)1/2. fσ stands for the orientational free energy (Forient) which describes the entropy loss when a preferred ordering exists, that is, fσ = Forient/(NkBT). These two terms account for the ideal contribution to the Helmholtz free energy. f(Ω) is the orientational distribution function. The last term of eq 1 depicts the mutual interaction between the molecules (Fconf), and the form of which in eq 1 is obtained when the higher virial coefficients are truncated. By analogy with this treatment, we divide the free energy into three parts. The difference is only the forms of Forient and Fconf when incorporating the flexibility and more complicated interactions. 1. Model. The wormlike chain model is frequently used to represent the semiflexible chain though it can represent molecules with a large extent of flexibility. Artificially, when the polymer is stretched into a linear configuration, we can take the model as a spherocylinder with the length equal to the contour length of the polymer, that is, l þ d = Mw/ML. Here, d is the diameter of the spherocylinder, l is the length of the cylinder, Mw is the average molecular weight and ML is the molecular weight per contour length. The stiffness of the wormlike chain is expressed in the term of the persistent length q, which is equal to half of the Kuhn length in this model. As a first order approximation, a polymer can be broken up into a set of smaller subchains. Though the wormlike chain model should have a continuous contour and the configurational free energy may depend on the flexibility distribution along the chain, the chain can be considered to be equivalent to m freely jointed rigid spherocylinders when the effect of the flexibility on the configurational free energy is concerned. This model is shown in Figure 1. Each subchain is modeled as a rigid spherocylinder and its length is equal to that of the Kuhn segment. When the contour length of the polymer is more than twice the persistent length, the number of subchain m is equal to the number of the Kuhn segment M. When the length of polymer is less than a Kuhn segment (M < 1), the chain is treated as a rigid rod (m = 1). The similar model was adopted by Yu and Liu.40 The varied phase behaviors exhibited by the LC materials are barely determined by the hard-core repulsive interaction. It is difficult to formulate a real potential function between polymers for the asymmetric, irregular shape and the complicated nature of mutual interactions which might contain electrostatic repulsion, dipole, multipole, isotropic, and anisotropic dispersion interactions, etc. The regular framework of the potential function of a spherocylinder model can be expressed as uðr12 , Ω1 , Ω2 Þ ¼ uhc ðr12 , Ω1 , Ω2 Þ þ upert ðr12 , Ω1 , Ω2 Þ ð2Þ and the reference hard-core discussed in this work takes the form ( ¥ r12 e σð^r12 , Ω1 , Ω2 Þ ð3Þ uhc ðr12 , Ω1 , Ω2 Þ ¼ 0 r12 > σð^r12 , Ω1 , Ω2 Þ where r12 and r12 are the intermolecular vector and its magnitude between the mass centers, and σ(^r12,Ω1,Ω2) is the contact 6461

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distance used to characterize the excluded volume. Due to the symmetry of the spherocylinder model, the intermolecular interaction is invariant to the inversion of the particle axis and thus the perturbative term can be expressed with even Legendre functions upert ðr12 , Ω1 , Ω2 Þ ( ε0 ε2 P2 ðcos γÞ σð^r12 , Ω1 , Ω2 Þ < r e ðlsc þ dÞ ¼ 0 others

ð4Þ g ¼

Here the perturbative term is supposed to have a square-well framework, therefore a minus is added to the expression indicating a potential well. ε0 in eq 4 stands for the isotropic interaction and ε2P2(cos γ) stands for the anisotropic part (i.e., the MaierSaupe interaction). The upper limit of the perturbative interaction is supposed to be a sphere with a diameter equal to the length of the spherocylinder, as shown in Figure 1. Such a model is vividly described as “convex peg in a round hole”.51 The more detailed description about the potential function can be directed to a recently published article by FrancoMeglar et al.52 The free energy of this model can be written as res F ¼ NkB Tf½lnðFΛ0 Þ 1 þ f σ g þ Fhc þ F pert

ð5Þ

Where Λ0 is a constant relating to the molecular property, Fres hc and Fpert are the Helmholtz free energy due to the hard core and van der Waals interaction, respectively. 2. Orientational Free Energy. The KS theory, 12,13 using the wormlike chain model and taking the average tangent vector along the chain as the preferred ordering direction, has been successfully extended to the calculation of the orientational free energy for the polymers with flexibility. Even if the Onsager trial function7 (OTF) is used to simplify the solution of the variational problem, the KS theory has an explicit expression only when the number of Kuhn segments is much larger than unity (almost random coil) or much smaller than unity (almost rigid rod). Different ways of handling the intermediate situations were proposed.4547 Dupre-Yang47 interpolation function provides a good approximation on both limits and also a smooth connection between them. Their expression is M f σ ðRÞ ¼ lnðRÞ 1 þ πeR þ ðR 1Þ 3 5 2M þ ln cosh ðR 1Þ 12 5

ð6Þ

where R is the variational parameter in the OTF and M is the number of the Kuhn segments in the wormlike chain. We also adopt this treatment in this work together with the OTF as the way to find the extreme value of the free energy functional rather than the conventional EulerLagrange equation. 3. Hard Core Contribution to the Configurational Free Energy. To conserve the packing fraction of the polymer, when m > 1, the length of the subchain should be 2 l ðm 1Þd 3 lsc ¼ m

CarnahanStarling equation 48 to represent the translational part, so the residual compressibility factor of the hard spherocylinder including the higher virial coefficient contribution is 4η 2η2 δ1 ð8Þ 1þ g Zsc ¼ 1 þ 4 ð1 ηÞ3 3 2 2 xsc 3x 2 sc

þ1

ð9Þ

where η = Nv0/V is the packing fraction of the system, x sc = lsc/d is the aspect ratio of the subchain, v0 is the molecular volume, and δ1 refers to the effect of the preferred orientation on the configurational free energy. In the isotropic phase δ1 = 1 and in the anisotropic phase 4 δ1 ¼ π

ZZ sin γf ðΩ1 Þ f ðΩ2 Þ dΩ1 dΩ2

ð10Þ

where γ is the angular separation of the axis. If the OTF is used, we have an explicit expression of R from eq 10: δ1 ¼

2I2 ð2RÞ sinh2 R

ð11Þ

where In is a modified Bessel function of the order n. In the ParsonsLee approximation, the sitesite correlation function of the hard spherocylinders is assumed to be equal to that of the hard spheres with the same packing fraction.9 If not so, each subchain can be considered as a linear tangent hard sphere chain so that the associating interaction between two spherocylinders can be treated as the interaction between the two terminal hard hemispheres.40 Both methods give the same result because the sitesite correlation function of the hard spheres used is only determined by the packing fraction, though the former method is much more artificial than the later one. As we know, the Chapman equation deviates from the MC simulation when the chain becomes longer.49 As a first-order approximation, the main reason for this is thought to be the neglect of the effect which is caused by the adjacent spheres on the sitesite correlation function of the two concerning spheres. Yu et al.50 developed an equation of state for the hard-sphere-chain (HSC) fluid in consideration of the “correlation hole”. This equation shows better precision covering a large range of the chain lengths when compared with other four equations. We employ Yu’s idea to modify the associating interaction between the two terminal hemispheres of the spherocylinders, so that the residual compressibility factor of the reference hardchain is 4η 2η2 δ1 res 1þ g Zhc ¼ m 4 ð1 ηÞ3 ð1 0:457η 2:104η2 þ 1:755η3 Þ ð1 ηÞ3 0:755ηð1 4:626η þ 6:321η2 Þ ðm 2Þ ð12Þ ð1 ηÞ3

ð7Þ

ðm 1Þ

Using the ParsonsLee approximation, 8,9 the translational contribution and the rotational contribution to the partition function can be decoupled into two parts. Lee9 used the 6462



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4 4 F11 ¼ ε0 Vpert ¼ ε0 πD3 ¼ ε0 πðx þ 1Þ3 d3 3 3 δ1 F12 ¼ ε0 hVexc iΩ1 , Ω2 ¼ 8vm ε0 1 þ g 4

The free energy of the hard-sphere chain contribution can be obtained as follows 2

res Fhc 4η 3η δ1 ¼m 1þ g NkB T 4 ð1 ηÞ2 "

þ ð1 mÞ ln η 2:755 lnð1 ηÞ " þ 0:755ð2 mÞ

# 0:406 0:097 þ 1 η ð1 ηÞ2

8:015 1:347 þ 6:321 lnð1 ηÞ 1 η ð1 ηÞ2

F14 ¼ ε2 hVexc P2 ðcos γÞiΩ1 , Ω2 g 9g 2 ¼ 8vm ε2 S2 þ δ1 δ2 4 32

ð13Þ

the perturbative potential function, the Helmholtz free energy can be calculated. The simplified treatment shows good agreements with simple fluids relying on the parameters selected, though the approximation itself is not sufficient. The complexity of the interactions between polymers makes it untraceable without simplification and the potential function adopted has brought in artificiality to some extent, therefore we still use the same approximation to test the roles of different parts of the perturbative interactions. What should also be emphasized is that the perturbative term in the free energy is based on the structure of the reference fluid when dealing with the simple fluid. In our theory, it means that f(Ω) should be first obtained by solving the hard core equation. As in the density functional theory,53 only the form of the perturbative free energy is retained and the phase transition conditions are determined by finding the extreme value of the whole free energy functional. The perturbative free energy of a spherocylinder with the firstorder approximation takes the general form ZZ ZZ m2 pert upert ðr12 , Ω1 , Ω2 Þ Fðr1 , Ω1 Þ Fðr2 , Ω2 Þ ¼ F 2 g12 ðr12 , Ω1 , Ω2 Þ dr1 dr2 dΩ1 dΩ2 ð14Þ where g12(r12, Ω1, Ω2) is the sitesite correlation function. When the conditions depicted above are used, after some mathematical refinement, the perturbative free energy of our model becomes Fpert F1 ¼m NkB T kB T ZZ m2 F ¼ ½ε0 þ ε2 P2 ðcos γÞðVpert Vexc Þf ðΩ1 Þ f ðΩ2 Þ dΩ1 dΩ2 2kB T

ð15Þ where Vexc is the excluded volume of the hard-core and Vpert is the range of the perturbative interaction, that is, Vexc

4 ¼ 2πlsc d2 þ πd3 þ 2lsc 2 d sin γ 3

ð20Þ

4 F13 ¼ ε2 Vpert hP2 ðcos γÞiΩ1 , Ω2 ¼ ε2 πðx þ 1Þ3 d3 S2 2 ð21Þ 3

#

4. Perturbative Contribution to the Configurational Free Energy. By a combination of the mean-field approximation and

ð19Þ

ð22Þ

)Here vm is the volume of the subchain; S2 is the order parameter of the system which is zero in the isotropic phase; δ2 is an integral of f(sin γ) like δ1, and in the isotropic phase δ2 = 1. The definitions of these terms are 1 1 vm ¼ πd3 þ πd2 lsc 6 4 S2 ¼ 1 δ2 ¼

3coth R 3 þ 2 R R

2½I2 ð2RÞ I6 ð2RÞ sinh2 R

ð23Þ ð24Þ ð25Þ

From eqs 1822, the perturbative free energy of our model can be expressed as ε0 ε2 2 pert 1 þ S2 ðb 1Þ F =ðNkB T Þ ¼ 4mη kT ε0 g ε2 g 9g δ 1 δ2 δ1 þ ð26Þ 4 32 ε0 4 where b¼

ðxsc þ 1Þ3 3x þ 1 2 sc

ð27Þ

An investigation of eqs 1922 highlights the fact that even an isotropic interaction may have an important impact on the IN phase transition due to the intense coupling between the perturbative interaction and the excluded volume. 5. Calculation of Phase Equilibrium. The compressibility factor, the osmotic pressure, and the chemical potential used in phase equilibrium calculations can be obtained from the Helmholtz free energy via Z¼

ð16Þ

η DF NkB T Dη

ð28Þ

Πv0 ¼ ηZ kB T

ð29Þ

After algebraic expansions, the integration can be calculated by parts

μ F ¼ þZ kB T NkB T

ð30Þ

F pert m2 F ¼ ðF11 F12 þ F13 F14 Þ kB T NkB T

The phase boundary concentrations depend on the extreme value of the free energy functional. Using OTF, it takes

Vpert

4 ¼ πðlsc þ dÞ3 3

ð17Þ

ð18Þ

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Figure 2. The reduced orientational free energy for chains with different numbers of Kuhn segments. The solid lines labeled with M represent the dependence of the orientational free energy on the chain flexibility, calculated by eq 6, and the dash line represents the rigid rod limit given by the Onsager theory.

Figure 3. The effect of chain flexibility on the phase boundary packing fractions for both isotropic phases and the nematic phases. The structural parameters used in calculation are ML = 740 g 3 mol1 3 nm1, q = 37 nm and d = 1.6 nm. The solid lines represent the results given by Dupre-Yang equation and the dash lines are given by the hard core part of eq 5.

the form DFaniso ¼0 DR

ð31Þ

With the common phase coexistence conditions, the following equations should be satisfied Πiso ¼ Πaniso

ð32Þ

μiso ¼ μaniso

ð33Þ

Equations 3133 can be solved to obtain the equilibrium compositions of isotropic and nematic phases.

’ RESULTS AND DISCUSSION The main improvement we make on the original work of Onsager is the incorporation of the flexibility and the perturbative interactions. When the chain is relatively rigid, the hard core equation degenerates to the Dupre-Yang treatment, and there are no more comments made on this situation. In the following part, the roles of each part of the modifications are first discussed in

Figure 4. The effect of varying the isotropic interactions on the IN phase equilibrium. The anisotropic interaction ε2 is set to be zero. The structural parameters used in calculations are ML = 740 g 3 mol1 3 nm1, q = 37 nm, and d = 1.6 nm. Circles, square, and triangles represent ε0/(kBT) = 0, 0.005, and 0.01, respectively. The solid and dashed lines represent the packing fractions of anisotropic and isotropic phases, respectively.

detail and then the equation is tested against the simulation and experimental data. 1. Influence of the Flexibility. The flexibility does have an impact on the formation of the anisotropic phase. In our model, the effects of both the orientational and configurational free energies are considered. In Figure 2, the reduced orientational free energy fσ is plotted against the variational parameter R, and polymers with different number of segments are shown. It is obvious that the treatment of the orientational entropy with the KS theory has a remarkable impact on this term. In the anisotropic phase, where R is relatively large, the deviations from Onsager’s rigid rod limit are too large to be neglected. On the other hand, its influence on the configurational free energy is marginal. In Figure 3, different computed results with the same molecular parameters are plotted against the number of the Kuhn segments. Phase separation is not strongly influenced by the appearance of the flexibility. The biphasic gap, which refers to the concentration difference between the isotropic phase and the anisotropic phase, has no significant difference between the two equations, but the phase boundary concentrations in both phases become lower in present theory. Nevertheless, this treatment is still important when the influence of the perturbative term is involved. 2. Influence of the Perturbative Interaction. In the framework of our theory, the form of the perturbative potential is supposed to be an attractive interaction. It has been proved that the combination of the attractive interaction to the spherocylinder model make it possible to describe more phase behaviors than IN phase transition, such as vaporisotropicanisotropic triple-point.21,54 The range of the perturbative interaction hardly has a significant impact on the concerning phase equilibrium phenomena, though it does affect others.52 In this section, the effect of the potential parameters is discussed. In Figure 4, the anisotropic part of the perturbative term is supposed to be zero, and only the isotropic interaction is considered. The phase boundary is significantly broadened as ε0/kBT increases, which is in accordance with the prediction from the theory and the well-established idea that even isotropic attractive interaction can enhance the degree alignment of the system and the first order character of the phase transition. In Figure 5, when the isotropic interaction is held to be constant, the 6464



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Table 1. the Structural Parameters of the Polymers parameters ML (g 3 mol1 3 nm1)

q (nm)

PHIC in toluene

740

37

1.25

PHIC in DCM

740

21

1.25

PBLG in DMF

1450

80

1.6

schizophyllan in water

2150

200

system

Figure 5. The effect of the anisotropic interactions on the IN phase equilibrium. The isotropic interaction ε0/(kBT) is set to be constant at 0.005. The solid and dashed lines represent packing fractions of the anisotropic and isotropic phases, respectively. Circles, squares, triangles, and crosses represent ε2/ε0 = 0, 0.1, 0.2, and 0.3, respectively. The structural parameters used in calculations are ML = 740 g 3 mol1 3 nm1, q = 37 nm, and d = 1.6 nm.

Figure 6. (a) Compressibility factor for chains of m = 2, 4, 8, 16 and (b) m = 51, 201 segments at different packing fractions. The solid lines are calculated with eq 34 and the dash lines are calculated with the Chapman equation. The symbols in both figures are MC or MD simulation results given by different authors.6164.

effect of the anisotropic part, which is well-known as the MaierSaupe interaction, is examined. As expected, the biphasic gap is enhanced in this case. What is more, the concentrations of both phases become lower as ε2 increases. We adopt this idea when comparing the theory with the experiments.

d (nm)

1.52.6

3. Freely Jointed Chain. When the persistent length q = d/2, the aspect ratio becomes zero. The orientation of the subchain disappears for its spherical symmetry and the whole molecule becomes a random coil. The compressibility factor of the equation for the hard core is

Z¼m

1 þ η þ η2 η3 ð1 ηÞ3

ð1 0:457η 2:104η2 þ 1:755η3 Þ ð1 ηÞ3 0:755ηð1 4:626η þ 6:321η2 Þ ðm 2Þ ð1 ηÞ3 ðm 1Þ

ð34Þ

This is the Yu equation.50 It is not coincident because the ParsonsLee approximation we use in this work is a scaling technique based on the hard sphere model, and its expression is derived from the CarnahanStarling equation which is also the reference model of the Yu equation. As discussed above, the Yu equation has a modification on the first-order thermodynamic perturbation so that it has a better performance than other equations as the length of the chain increases. In Figure 6 the results of eq 34 are plotted with the simulation data, and the Chapman equation which has no modification on the “correlation hole” is used as a comparison. The curves are labeled with m. It can be seen from the figure that good agreements are obtained by both equations when the length of the chain is relatively short. When m = 16, the result given by the Chapman equation is apparently higher than the simulation result, and the deviation becomes larger as the chain grows longer. In contrast, the Yu equation reproduces the compressibility factors very well for chains with different lengths. 4. Comparing with Experimental Data. The dependence of the phase boundary concentrations on the Kuhn segments can be measured from experiments on the same polymer with different molecular weights. To apply the theory to real systems, three parameters of the molecule are required, the molecular weight Mw of the polymer together with the molecular weight per unit contour length ML, the persistent length q, and the spherocylinder diameter d. In some hard-core theories, the diameter, sometimes together with the persistent length, is chosen as the adjustable parameter so that the theory can give the best agreement with the experiment, This treatment is not very convincing.40,55 The theory is usually used to predict the concentration when the anisotropic phase starts to form. In this condition, only the boundary concentration of the isotropic phase is concerned. Because the Onsager-like theory gives a good qualitative trend of the phase diagram, this objective can be easily achieved if some parameters are regressed from the experimental data, but the result of the anisotropic phase may deviate from the experiment severely. In this section, we compare our theoretical with the experimental phase boundary concentrations for several semiflexible polymer solutions. These 6465



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Figure 7. The chain length dependence of ηi and ηa for PHIC in toluene at 25 C. Circles and squares are the experimental data56,57 of the two phases, respectively. The solid lines are the results given by eq 5 with the parameters d = 1.25 nm, ε0/(kBT) = 3 103, ε2/(kBT) = 7.5 104 and the dash lines are the results given by the DupreYang theory with the same diameter.

Figure 9. The chain length dependence of ηi and ηa for PBLG in DMF. Circles and squares are the experimental data58 of the two phases, respectively. The solid lines are the results given by eq 5 with the parameters d = 1.60 nm, ε0/(kBT) = 2.2 103, ε2/(kBT) = 2 104 and the dash lines are the results given by the Dupre-Yang theory with the same diameter.

Figure 8. The chain length dependence of ηi and ηa for PHIC in DCM at 20 C. Circles and squares are the experimental data56,57 of the two phases, respectively. The solid lines are the results given by eq 5 with the parameters d = 1.25 nm, ε0/(kBT) = 1.8 102, ε2/(kBT) = 5 104 and the dash lines are the results given by the Dupre-Yang theory with the same diameter.

Figure 10. The chain length dependence of ηi and ηa for schizophyllan in water at 25 C. Circles and squares are the experimental data55 of the two phases, respectively. The solid lines are the results given by eq 5 with the parameters d = 2.10 nm, ε0/(kBT) = 1.1 103, ε2/(kBT) = 2.2 104, and the dash lines are the results given by the DupreYang theory with the same diameter.

systems include poly(hexyl isocyanate) (PHIC) in dichloromethane (DCM) and toluene,56,57 poly-γ-benzyl-L-glutamate (PBLG) in dimethylformamide (DMF),58 and schizophyllan in water.55 The obtained structural parameters are listed in Table 1. Though schizophyllan in water is a cholesteric liquid crystal rather than a nematic one, the free energy difference between the nematic phase and the cholesteric phase is very small and they are always treated as the same in the calculation. If the diameter d is chosen as a regression parameter, only isotropic interaction parameter itself can give a relatively precise description of the phase behavior. In the present theory, we uses the diameter d measured from the partial specific volume, the persistent length q obtained from the measurement of the intrinsic viscosity, and the molecular weight per contour length ML regressed from the experimental data. It is worthwhile noting that under framework of our theory, the effect of the perturbative interaction term is trivial when the molecules are short (i.e., the length is shorter than a persistent length) so that the deviation from the experiment is inevitable if the diameter d is not adjustable. Results when the diameter d is used as one of adjustable

parameters can be found in the Supporting Information. Both the isotropic and the anisotropic perturbative terms in the equation are retained. What should be emphasized is that the perturbative interaction rather than the attractive is used in this work. When other forms of potential functions are used, like the Lennard-Jones potential function, the interaction itself may not totally be an attractive one. Moreover, in our comparison with the experiments, though the whole perturbative interaction is still an attractive one, the anisotropic interaction takes a repulsive form in some cases (socalled “square shoulder”). This may correspond to some realities, but more probably it is caused by the defect of the model and the regression method we choose. If we have a physical significance imposed on this result, the answer might be misleading. Yu’s theory is a good work which also gives us the inspiration of this work. Yu’s theory incorporates the flexibility in the configurational free energy based on a scaled particle theory, but this change itself cannot improve the precision of the equation too much. Yu’s theory has almost the same results as those of DupreYang theory. Therefore, we think a comparison with the DupreYang equation 6466


Industrial & Engineering Chemistry Research is persuasive to demonstrate the improvement of our theory over these kinds of theories. In Figures 710, the chain length dependences of equilibrium concentrations for isotropic and nematic phases are presented. From the figures one can see that the present theory predicts the IN phase equilibria well for the four systems studied. In contrast, the DupreYang theory substantially underestimates the equilibrium concentrations of both phases for PHIC in toluene at 25 C and predicts a narrower biphasic gap for the systems of PHIC in DCM, PBLG in DMF, and schizophyllan in water when compared with the experimental data.5558 These results indicate that the inclusion of perturbative interaction to the Helmholtz free energy can improve the prediction of the IN phase equilibria. For the polydisperse system, the average molecular weight is used so that a problem concerning the polydisperse is eschewed, which also implies the theory itself is more practical in the lowlevel polydisperse. Some methods59,60 to extend the Onsager-like theories to bidisperse, polydisperse systems and mixtures have been proposed. Since the framework of our theory is also Onsager-like, the methods can be applied in the same way to the polydisperse systems.

’ CONCLUSIONS Based on the thermodynamic perturbation theory of hardsphere-chain fluid, the Onsager-like theory is extended to a more realistic model incorporating the molecular flexibility and the perturbative interactions. In the new equation, the entropy loss due to orientation is estimated by the KS theory where explicit expression is given by the DupreYang interpolation equation. With regard to the configurational free energy, the polymer is envisioned as a series of subchains. The ParsonsLee approximation is used to account for the higher virial coefficients of the subchains and Yu equation is adopted to modify the defect of the first order approximation in dealing with the associating points. An analytical expression of the perturbative contribution to the configurational free energy is obtained by employing the “square peg in a round hole” potential function and the mean-field approximation. The equation reduces to the Dupre-Yang equation when the stiffness of the molecule is high. On the other hand, when it approaches the limit of the random coil, an equation with better performance for the long chains is obtained. As for the real semiflexible polymers, the agreement between theory and experimental data is much better than the previous studies when two parameters in the equation are regressed from the experimental data. In the one-component systems, the two parameters have an exact physical significance that one stands for the isotropic perturbative interaction and the other stands for the anisotropic interaction (the MaierSaupe interaction). What should be paid attention to is that, though the theory gives a resolution to the IN transition for the random coils, the validity of the equation in this condition is not verified. ’ ASSOCIATED CONTENT

bS

Supporting Information. The integrating process of the perturbative free energy and additional figures when the diameter is used as an adjustable parameter regressed from the experimental data. This material is available free of charge via the Internet at http://pubs.acs.org.

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’ AUTHOR INFORMATION Corresponding Author

* E-mail: [email protected].

’ ACKNOWLEDGMENT We would like to make an acknowledgement to Yong-Jun Du for her effort in the preparation of the manuscript. This work is supported by the National Natural Science Foundation of China (No. 20876083 and No. 20736003), the Major State Basic Research Program of China (No. 2009CB623404), and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20100002110024). ’ NOMENCLATURE b = algebraic expression of x d = diameter of the spherocylinder, nm Fconf = configurational free energy, J Fid = ideal free energy, J Forient = oriental free energy, J Fpert = perturbative free energy, J Fres = residual free energy, J Fres hc = residual free energy of the hard core, J f(Ω) = orientational distribution function fσ = reduced orientational free energy g = algebraic expression of x g12 = pair correlation function In = modified Bessel function of the order n kB = Boltzmann constant, J/(mol 3 K) l = length of the cylinder, nm lsc = length of the cylinder in the subchain, nm ML = molecular weight per contour length, g/(mol 3 nm) Mw = average molecular weight, g/mol N = number of the molecules in the system q = persistent length, nm r12 = intermolecular vector between the mass centers, nm r12 = magnitude of the intermolecular vector, nm S2 = order parameter uhc = hard core part of the potential function, J upert = perturbative part of the potential function, J Vexc = excluded volume of the subchain, nm3 Vpert = range of the mutual perturbative interaction of the subchain, nm3 v0 = volume of the molecule, nm3 vm = volume of the subchain, nm3 x = aspect ratio of the spherocylinder Z = compressibility factor Zres hc = residual compressibility factor of the hard core Greek Symbols

r = variation parameter of OTF γ = angular separation of the axis, rad δ1 = algebraic expression of R δ2 = algebraic expression of R ε0 = isotropic perturbative interaction, J ε2P2(cos γ) = anisotropic perturbative interaction, J η = packing fraction of the system Λ0 = constant relating to the molecular property μ = chemical potential of a molecule, J/mol ν = νtνr = translational and rotational contribution to partition function 6467


Industrial & Engineering Chemistry Research Π = osmotic pressure, Pa F = number density of the molecules in the system, nm3 σ = contact distance of the two spherocylinder, nm Ω = orientational angle, rad

’ SUBSCRIPT aniso and a = anisotropic iso and i = isotropic ’ REFERENCES (1) Flory, P. J. Phase equilibria in solutions of rodlike particles. Proc. R. Soc. London, Ser. A 1956, 234 (1196), 73–89. (2) Flory, P. J. Molecular theory of liquid crystals. Adv. Polym. Sci. 1984, 59, 1–36. (3) Flory, P. I. Thermodynamics of high polymer solutions. J. Chem. Phys. 1942, 10 (1), 51–61. (4) Huggins, M. L. Solutions of long chain compounds. J. Chem. Phys. 1941, 9 (5), 440–440. (5) Flory, P. J.; Ronca, G. Theory of systems of rodlike particles. 1. Athermal systems. Mol. Cryst. Liq. Cryst. 1979, 54 (34), 289–309. (6) Flory, P. J.; Ronca, G. Theory of systems of rodlike particles. 2. Thermotropic systems with orientation-dependent interactions. Mol. Cryst. Liq. Cryst. 1979, 54 (34), 311–330. (7) Onsager, L. The effects of shape on the interaction of colloidal particles. Ann. N.Y. Acad. Sci. 1949, 51 (4), 627–659. (8) Parsons, J. D. Nematic ordering in a system of rods. Phys. Rev. A 1979, 19 (3), 1225–1230. (9) Lee, S. D.; Meyer, R. B. Computations of the phase equilibrium, elastic constants, and viscosities of a hard-rod nematic liquid crystal. J. Chem. Phys. 1986, 84 (6), 3443–3448. (10) Cotter, M. A. Hard sphero-cylinders in an anisotropic mean field—Simple model for a nematic liquid crystal. J. Chem. Phys. 1977, 66 (3), 1098–1106. (11) Tjiptomargo, B.; Evans, G. T. The Onsager theory of the isotropic-nematic liquid crystal transition—Incorporation of the higher virial coefficients. J. Chem. Phys. 1990, 93 (6), 4254–4265. (12) Khokhlov, A. R.; Semenov, A. N. Liquid crystalline ordering in the solution of long persistent chains. Phys. A 1981, 108 (23), 546–556. (13) Khokhlov, A. R.; Semenov, A. N. Liquid crystalline ordering in the solution of partially flexible macromolecules. Phys. A 1982, 112 (3), 605–614. (14) Maier, W.; Saupe, A. Eine einfache molekulare theorie des nematischen kristallinflussigen zustandes. Z. Naturf. A 1958, 13 (7), 564–566. (15) Maier, W.; Saupe, A. Eine einfache molekular statistische theorie der nematischen kristallinflussigen phase. 1. Z. Naturforsch. A 1959, 14 (10), 882–889. (16) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University: London, 1986. (17) Cotter, M. A. Generalized van der Waals theory of nematic liquid crystals—Alternative formulation. J. Chem. Phys. 1977, 66 (10), 4710–4711. (18) Bolhuis, P. G.; Stroobants, A.; Frenkel, D.; Lekkerkerker, H. N. W. Numerical study of the phase behavior of rodlike colloids with attractive interactions. J. Chem. Phys. 1997, 107 (5), 1551–1564. (19) Varga, S.; Williamson, D. C.; Szalai, I. Square well fluid based decoupling approximation for system of hard nonspherical particles with spherical square wells. Mol. Phys. 1999, 96 (11), 1695–1703. (20) Williamson, D. C.; del Rio, F. The isotropic-nematic phase transition in a fluid of square well spherocylinders. J. Chem. Phys. 1998, 109 (11), 4675–4686. (21) Williamson, D. C.; Guevara, Y. Deviation from corresponding states for a fluid of square well spherocylinders. J. Phys. Chem. B 1999, 103 (35), 7522–7530.

ARTICLE

(22) de Gennes, P. G. Short range order effects in the isotropic phase of nematics and cholesterics. Mol. Cryst. Liq. Cryst. 1971, 12, 193–214. (23) Wertheim, M. S. Fluids with highly directional attractive forces. 1. Statistical thermodynamics. J. Stat. Phys. 1984, 35 (12), 19–34. (24) Wertheim, M. S. Fluids with highly directional forces. 2. Thermodynamic perturbation theory and integral-equations. J. Stat. Phys. 1984, 35 (12), 35–47. (25) Wertheim, M. S. Fluids with highly directional forces. 3. Multiple attraction sites. J. Stat. Phys. 1986, 42 (34), 459–476. (26) Wertheim, M. S. Fluids with highly directional attractive forces. 4. Equilibrium polymerization. J. Stat. Phys. 1986, 42 (34), 477–492. (27) Wertheim, M. S. Thermodynamic perturbation-theory of polymerization. J. Chem. Phys. 1987, 87 (12), 7323–7331. (28) Chapman, W. G.; Jackson, G.; Gubbins, K. E. Phase-equilibria of associating fluids chain molecues with multiple bonding sites. Mol. Phys. 1988, 65 (5), 1057–1079. (29) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT-equation of state solution model for associating fluids. Fluid Phase Equilib. 1989, 52, 31–38. (30) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New reference equation of state for associating liquids. Ind. Eng. Chem. Res. 1990, 29 (8), 1709–1721. (31) Ghonasgi, D.; Chapman, W. G. A new equation of state for hard chain molecules. J. Chem. Phys. 1994, 100 (9), 6633–6639. (32) Ghonasgi, D.; Chapman, W. G. Prediction of the properties of model polymer—Solutions and blends. AIChE J. 1994, 40 (5), 878–887. (33) Movahed, H. B.; Hidalgo, R. C.; Sullivan, D. E. Phase transitions of semiflexible hard-sphere chain liquids. Phys. Rev. E 2006, 73 (3), 032701. (34) Williamson, D. C.; Jackson, G. Liquid crystalline phase behavior in systems of hard-sphere chains. J. Chem. Phys. 1998, 108 (24), 10294–10302. (35) Williamson, D. C.; Jackson, G. Excluded-volume for a pair of linear-chains of tangent hard-spheres with an arbitrary relative orientation. Mol. Phys. 1995, 86 (4), 819–836. (36) Hino, T.; Prausnitz, J. M. Hard-sphere-chain equations of state for lyotropic polymer liquid crystals. Polymer 1999, 40 (5), 1241–1250. (37) Burgess, W. A.; Zhuang, M. S.; Hu, Y.; Hurt, R. H.; Thies, M. C. SAFT-LC: An equation of state for predicting liquid crystalline phase behavior in carbonaceous pitches. Ind. Eng. Chem. Res. 2007, 46, 7018–7026. (38) Hu, Y.; Liu, H. L.; Prausnitz, J. M. Equation of state for fluids containing chainlike molecules. J. Chem. Phys. 1996, 104 (1), 396–404. (39) Vega, C.; Lago, S. Isotropic-nematic transition of hard polar and nonpolar molecules. J. Chem. Phys. 1994, 100 (9), 6727–6737. (40) Yu, M.; Liu, Q. An equation of state for polymer molecules: From flexible chains to rigid rods. Macromolecules 1996, 29 (21), 6928–6932. (41) Jaffer, K. M.; Opps, S. B.; Sullivan, D. E.; Nickel, B. G.; Mederos, L. The nematic-isotropic phase transition in semiflexible fused hardsphere chain fluids. J. Chem. Phys. 2001, 114 (7), 3314–3324. (42) Jiang, T.; Wu, J. Z. Isotropic-nematic phase transition in athermal solutions of rodcoil diblock copolymers. J. Chem. Phys. 2007, 127 (3), 9. (43) Vroege, G. J.; Lekkerkerker, H. N. W. Phase-transitions in lyotropic colloidal and polymer liquid crystals. Rep. Prog. Phys. 1992, 55 (8), 1241–1309. (44) Franco-Melgar, M.; Haslam, A. J.; Jackson, G. A generalization of the Onsager trial function approach: Describing nematic liquid crystals with an algebraic equation of state. Mol. Phys. 2008, 106 (5), 649–678. (45) Odijk, T. Theory of lyotropic polymer liquid crystals. Macromolecules 1986, 19 (9), 2314–2329. (46) Hentschke, R. Equation of state for persistent flexible liquid crystal polymers—Comparison with poly(gamma-benzyl-L-glutamate) in dimethylformamide. Macromolecules 1990, 23 (4), 1192–1196. (47) Dupre, D. B.; Yang, S. J. Liquid crystalline I properties of solutions of persistent polymer chains. J. Chem. Phys. 1991, 94 (11), 7466–7477. (48) Carnahan, N. F.; Starling, K. E. Equation of state for nonattracting rigid spheres. J. Chem. Phys. 1969, 51 (2), 635–636. 6468



ARTICLE

(49) Chang, J.; Sandler, S. I. An equation of state for the hard-sphere chain fluid—Theory and Monte-Carlo simulation. Chem. Eng. Sci. 1994, 49 (17), 2777–2791. (50) Yu, Y. X.; Lu, J. F.; Tong, J. S.; Li, Y. G. Equation of state for hardsphere chain molecules. Fluid Phase Equilib. 1994, 102 (2), 159–172. (51) Williamson, D. C. The “convex peg” model: The long range approximation. Mol. Phys. 1998, 95 (2), 319–329. (52) Franco-Melgar, M.; Haslam, A. J.; Jackson, G. Advances in generalised van der Waals approaches for the isotropic-nematic fluid phase equilibria of thermotropic liquid crystals—An algebraic equation of state for attractive anisotropic particles with the Onsager trial function. Mol. Phys. 2009, 107 (22), 2329–2358. (53) Yu, Y.-X. A novel weighted density functional theory for adsorption, fluidsolid interfacial tension and disjoining properties of simple films on planar solid surfaces. J. Chem. Phys. 2009, 131, 024704. (54) Garcia, E.; Williamson, D. C.; Martinez-Richa, A. Effects of molecular geometry on liquid crystalline phase behaviour: Isotropicnematic transition. Mol. Phys. 2000, 98 (3), 179–192. (55) Sato, T.; Teramoto, A. Isotropic liquid crystal phase equilibrium semiflexible polymer solutions—Application of scaled particle theory. Mol. Cryst. Liq. Cryst. 1990, 178, 143–155. (56) Itou, T.; Sato, T.; Teramoto, A.; Aharoni, S. M. Thermodynamic properties of concentrated solutions of poly(hexyl isocyanate), a liquid-crystalline polymer. Polym. J. 1988, 20 (11), 1049–1058. (57) Itou, T.; Teramoto, A. Isotropic liquid crystal phase equilibrium solutions of semiflexible polymerspoly(hexyl isocyanate). Macromolecules 1988, 21 (7), 2225–2230. (58) Kubo, K.; Ogino, K. Comparison of osmotic pressures for the poly(gamma-benzyl-L-glutamate) solutions with the theories for a system of hard spherocylinders. Mol. Cryst. Liq. Cryst. 1979, 53 (34), 207–227. (59) Lekkerkerker, H. N. W.; Coulon, P.; van der Haegen, R.; Deblieck, R. On the isotropic liquid crystal phase separation in a solution of rodlike particles of different lengths. J. Chem. Phys. 1984, 80, 3427–3433. (60) Odijk, T.; Lekkerkerker, H. N. W. Theory of the isotropic liquid crystal phase separation for a solution of bidisperse rodlike macromolecules. J. Phys. Chem. 1985, 89, 2090–2096. (61) Dickman, R.; Hall, C. K. High-density Monte Carlo simulations of chain molecules—Bulk equation of state and density profile near walls. J. Chem. Phys. 1988, 89 (5), 3168–3174. (62) Denlinger, M. A.; Hall, C. K. Molecular dynamics simulation results for the pressure of hard-chain fluids. Mol. Phys. 1990, 71 (3), 541–559. (63) Amos, M. D.; Jackson, G. BHS theory and computer simulations of linear heternuclear triatomic hard-sphere molecules. Mol. Phys. 1991, 74 (1), 191–210. (64) Gao, J.; Weiner, J. H. Contribution of covalent bond force to pressure in polymer melts. J. Chem. Phys. 1989, 91 (5), 3168–3173.

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