In 1997, the first cloned animals were generated [12]. Cytogenetic analyses have shown that karyotypic aberrations occur in cloned embryos during the first mitotic cleavage. Epigenetic errors in cloned animals are also argued to be the major reason for the limited success rates of cloned animal births. However, whether genome-wide chromatin remodeling during nuclear reprogramming causes DNA damage, or whether defects in DNA repair cause the inefficiency of cloned animal births have not been investigated. Recently, it was reported that double strand break (DSB)–mediated chromatin remodeling regulates transcription, and that Parp1 is critically involved in this process. Various roles for Parp1 have been described, including in the chromatin remodeling involved in transcriptional regulation [2,3]. The roles of Parp1 in NT-embryo development in the contexts of DNA repair and chromatin remodeling was examined. To do this, we used *Parp1*-null mutant cells as recipient oocytes and as a source of donor nuclei.

We observed that the activation process in NT eggs was enhanced in *Parp1*–null NT embryos, although some genomic instability was observed (Osada T., unpublished data). Dynamic changes in histone acetylation and methylation were induced under *Parp-1* deficiency in NT embryos (Figure 3). The lack of Parp1 may thus facilitate chromatin condensation or transcriptional silencing. Parp1 is involved in both the regulation of DNA strand break repair and the epigenetic control of gene expression. Our findings suggest that Parp1 is important for chromatin remodeling possibly through histone modification during the nuclear reprogramming of NT embryos. The phosphorylation of histone H2AX at DSBs is believed to be crucial for the recognition and repair of DNA damage. The foci-like presence of H2AX in the NT embryos implied that DNA repair was taking place in the PNN. H2AX is a direct target of ATM kinase, which is expressed at the spindle of MII oocytes, and Parp1 interacts with ATM. The prolongation of the H2AX foci in the *Parp1-/-* NT embryos could therefore be related to a delay in repairing DNA strand breaks, as is the case with *Parp1-/-* MEF and ES cells, which show prolonged H2AX foci after DNA damage induced by neocartinostatin. In *Parp1-/-* NT embryos, mitotic arrest of embryogenesis at the 2-4-cell stage was frequently observed, accompanied by polynucleated blastomeres. Since Parp1 is located in centrosomes as well as in nuclei, the lack of Parp1 may have disturbed the normal cell division cycle and the synchronous pattern of cell division among the blastomeres. Transient Parp-1 functional inhibition may be therefore useful to improve the efficiency of NT by modulating the dynamic organization of chromatin without causing genomic instability.

#### **4. Implication towards human diseases**

PARylation has been known to possess important roles in carcinogenesis. PARP mainly affects DNA surveillance system in the cells, which deficit increases the risk of carcinogenesis. In addition, recent study showed a functional relationship between PARP and MAPK signaling, known as a major intracellular signaling of transformation into cancer cells. Although morphological change of nuclear envelope has been known in cancer cells, the molecular mechanism of the processes is not well understood. Recent studies revealed molecular signaling of laminar formation is altered during carcinogenesis. Our study revealed that PARylation is involved in these molecular events. Oocytes are capable of being isolated without other cell types by adding hyarulonidase, which is far easier compared with the difficulty in homogenous isolation of purified cancer cells. Our data suggest oocytes provide with a unique biological window for elucidating the mechanism of PARylation in carcinogenesis. Further comparative analysis between cancer cells and oocytes may highlight the uncovered mechanisms underlying the carcinogenesis processes.

**Figure 3. The scheme of histone modification and the effects of** *Parp1* **deficiency during the first cleavage of NT embryogenesis.** Histone modifications of the one-cell reconstructed oocytes were indicated in *Parp+/+*, and those which *Parp1* deficiency influences were indicated in *Parp-/-* in red.

We also described here that PARylation is a novel target of anti-conception. Inhibition of PARylation may easy to be handled, because exposure of an inhibitor to oocytes is effective enough to stop fertilization. Further extensive analysis should be carried out regarding the optimized dose of inhibitors and non-toxic dose on other tissues. Biological phenomena, in which PARylation is involved, are variable and its inhibitors could serve as possible pharmaceutical targets including inflammation or brain injury as well as carcinogenesis and reproduction as discussed in this chapter. Further basic investigation of the roles of PARylation in the cells will broaden the view for the understanding the embryogenesis and proof of mechanism of human diseases for drug discovery.

## **Author details**

328 Polymerization

processes.

Nuclear transfer

and MAPK signaling, known as a major intracellular signaling of transformation into cancer cells. Although morphological change of nuclear envelope has been known in cancer cells, the molecular mechanism of the processes is not well understood. Recent studies revealed molecular signaling of laminar formation is altered during carcinogenesis. Our study revealed that PARylation is involved in these molecular events. Oocytes are capable of being isolated without other cell types by adding hyarulonidase, which is far easier compared with the difficulty in homogenous isolation of purified cancer cells. Our data suggest oocytes provide with a unique biological window for elucidating the mechanism of PARylation in carcinogenesis. Further comparative analysis between cancer cells and oocytes may highlight the uncovered mechanisms underlying the carcinogenesis

PARP+/+ cloned one-cell embryo

↑ (after activation) H4K12 → (before/after activation)

↑(after activation) H3K9 di-, tri-, ↓ (after activation, non-asymetric)

Acetylation H3 ↓ (before activation)

Methylation H3K4 mono-, ↓(before activation)

Acetylation H3 ↓ (before activation)

Methylation H3K4 mono-, ↓(before activation)

H3K27 tri-, ↓(after activation) Phosphorylation H2AX ↑(After activation) Blastocyst Formation

No blastocyst Formation

PARP-/- cloned one-cell embryo

**↓ (after activation)** H4K12 → (before/after activation)

**↓(after activation)**  H3K9 di-, tri-, ↓ (after activation, non-asymetric) H3K27 tri-, **→(after activation)** Phosphorylation H2AX **→(After activation)**

**Figure 3. The scheme of histone modification and the effects of** *Parp1* **deficiency during the first cleavage of NT embryogenesis.** Histone modifications of the one-cell reconstructed oocytes were indicated in *Parp+/+*, and those which *Parp1* deficiency influences were indicated in *Parp-/-* in red.

Tomoharu Osada and Mitsuko Masutani *Mitsubishi Chemical Medience Corporation, Kashima Laboratory, Japan Division of Genome Stability Research, National Cancer Center Research Institute, Japan*

## **Acknowledgement**

We thank Dr. Ryuzo Yanagimachi and Dr. Takashi Sugimura for giving informative insight throughout the study. This study is partly supported by a Grand-in Aid for the Third Comprehensive 10-Year Strategy for Cancer Control from the Ministry of Health, Labor and Welfare of Japan.

## **5. References**

**Section 4**

## **Polymer Physics**

330 Polymerization

[9] Nakamura T, Arai Y, Umehara H, Masuhara M, Kimura T, et al (2007) PGC/Stella protects against demethylation in early embryogenesis. Nat. Cell Biol. 9: 64-71. [10] Reale A, Matteis GD, Galleazzi G, Zampieri M, Caiafa P, 2005 Modulation of DNMT1

[11] Quenet D, Gasser V, Fouillen L, Cammas F, Sanglier-Cianferani S, et al (2008) The histone subcode: poly(ADP-ribose) polymerase-1 (Parp-1) and Parp-2 control cell differentiation by regulating the transfcriptional intermediary factor TIF beta and

[12] Wilmut I, Schnieke AE, McWhir J, King AJ. Campbell KH (1997) Viable offspring

activity by ADP-ribosylation. Oncogene 24: 13-19.

heterochromatin protein HP1alpha. FASEB J. 22: 3853-3865.

derived from fetal and adult mammalian cells. Nature 385:810-813.

**Chapter 15**

## **Two-Photon Polymerization Fabrication of Doped Microstructures**

Daniel S. Correa, Leonardo De Boni, Adriano J. G. Otuka, Vinicius Tribuzi and Cleber R. Mendonça

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/36061

## **1. Introduction**

Scientists and engineers have sought to design more compact and efficient devices by means of microfabrication techniques. Examples of microfabrication processes include lithography, chemical vapor deposition, sol-gel, dry etching, among others. Although every technique has its own specificities and advantages, most of them are multi-step processes and demand long time of fabrication. Because of such limitations, and also because of the small range of materials that can be used in these techniques, there has been extensive search for alternative microdevice fabrication methods. A new microfabrication technique, called two-photon polymerization (2PP), emerged in 1997 opening a wider range of material possibilities with the advantage of allowing tridimensional fabrication. In this technique, Maruo et al (Maruo, et al., 1997) employed a laser-based-apparatus to fabricate three-dimensional polymeric microstructures with no topological constrains, high penetration depth without surface modifications and resolution bellow the diffraction limit. This technique usually employs femtosecond laser pulses to promote two-photon absorption (2PA) of a photosensitive molecule dissolved into the bulk of an unpolymerized resin, which creates a radical and triggers polymerization in a confined spatial region.

In a 2PA transition, an atom or molecule is taken to an excited state by simultaneously absorbing two photons in a single quantum event. Considering that 2PA is difficult to be attained by using conventional (low intensity) light sources, normally pulsed laser light is employed. Owing to the nonlinear nature of the 2PA, it displays a quadratic dependence to the light intensity, providing high spatial resolution and low light scattering. As an outcome for a polymerization relied on 2PA, chemical reaction takes place in a tiny focal volume, allowing the fabrication of tridimensional polymeric structures with resolution below the diffraction limit. For instance, two-photon polymerization has already been applied to fabri-

© 2012 Mendonça et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Mendonça et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

cate micro and nanodevices for a wide variety of applications (Kawata, et al., 2001, LaFratta, et al., 2007, Lee, et al., 2008, Marder, et al., 2007, Sun and Kawata, 2004).

To further enhance the technological potentiality of two-photon polymerization (2PP) to fabricate microstructures, one can dope the basic resin with dyes, nanoparticles, special polymers, and other materials. Such methodology leads to the fabrication of active microstructures with chemical, biological or optical properties, with potential applications in, for instance, optical data storage, photonic crystals, fluorescent and bioactive microstructures (Correa, et al.,2012, Correa, et al.,2009, Cumpston, et al.,1999, Drakakis, et al.,2006, Farsari, et al.,2008, Liska, et al.,2007, Mendonca, et al.,2007, Ovsianikov, et al.,2007, Serbin, et al.,2004, Sun, et al.,2001, Takeyasu, et al.,2008, Tayalia, et al.,2008). In this chapter, we will discuss theoretical and experimental aspects of the two-photon polymerization process, and will review the last advances in the fabrication of doped microstructures. Such microstructures present unique optical, electrical or biological properties (according to the dopant characteristics), desired for technological applications.

## **2. Two-photon absorption: Fundamentals**

Two-photon absorption is a process related to the change in the electronic state of an atom or molecule, which is excited from a lower energy level *n* (ground state, in the most of cases) to a higher electronic state *n* 1 , absorbing simultaneously two-photon of same (called "degenerate process") or different energies (called "nondegenerate process"). An intermediate virtual state is created from the interaction between the electromagnetic field and matter. Two-photon absorption was theoretically predicted by Maria Göppert-Mayer in 1931 during her doctoral thesis (Göppert-Mayer,1931). Only after thirty years, the experimental observation of the phenomenon became possible, thanks to the invention of the LASER. The first experimental verification of 2PA was confirmed by Kaiser et al, who detected the fluorescence signal emitted by a europium-doped crystal (Kaiser and Garrett,1961), whose electrons were excited via a two-photon transition. Furthermore, this effect was experimentally proved by Isaac Abella (Abella,1962) in 1962, using Cesium vapor as the two-photon absorbing material. Nowadays, two-photon absorption processes have been extensively studied because of the enormous number of technological applications, such as, multiphoton optical limiters, multiphoton fluorescence spectroscopy, as well as multiphoton polymerization. The importance of multi-photon absorption studies is not only restricted to technological applications, but also lies in basic research, which enables to characterize and gather information about different materials. In this field, the quantification of such effect can provide insights on the molecular geometry and electronic charge distribution of an excited molecule. Additionally, two-photon absorption measurements provide properties (symmetry and energy) of some electronic states that can not be seen using traditional spectroscopic experiments.

To promote an excitation by absorption of two photons, the sum of the energies of each isolated photon need to match the energy of the electronic transition, as illustrated in Fig. 1, for degenerated and nondegenerated cases. Such absorption process is classified as a simultaneous event. According to the uncertainty principle of Heisenberg, however, photons that participate in the transition can have a time interval given by h/∆E, where ∆E is the difference of energy between the material´s virtual state and the closest real state, and h is the Planck constant. In other words, the absorption of two photons does not need to be totally instantaneous, once it occurs in an interval given by the Heisenberg principle. Therefore the difference between the virtual and closest real state depends on the energy of the first photon that arrives in the molecule or atom. It means that if the photon presents energy close to the one between the states involved in the transition, according to the uncertainty principle, the 2PA probability will be enhanced.

334 Polymerization

cate micro and nanodevices for a wide variety of applications (Kawata, et al., 2001, LaFratta,

To further enhance the technological potentiality of two-photon polymerization (2PP) to fabricate microstructures, one can dope the basic resin with dyes, nanoparticles, special polymers, and other materials. Such methodology leads to the fabrication of active microstructures with chemical, biological or optical properties, with potential applications in, for instance, optical data storage, photonic crystals, fluorescent and bioactive microstructures (Correa, et al.,2012, Correa, et al.,2009, Cumpston, et al.,1999, Drakakis, et al.,2006, Farsari, et al.,2008, Liska, et al.,2007, Mendonca, et al.,2007, Ovsianikov, et al.,2007, Serbin, et al.,2004, Sun, et al.,2001, Takeyasu, et al.,2008, Tayalia, et al.,2008). In this chapter, we will discuss theoretical and experimental aspects of the two-photon polymerization process, and will review the last advances in the fabrication of doped microstructures. Such microstructures present unique optical, electrical or biological properties (according to the dopant character-

Two-photon absorption is a process related to the change in the electronic state of an atom or molecule, which is excited from a lower energy level *n* (ground state, in the most of cases) to a higher electronic state *n* 1 , absorbing simultaneously two-photon of same (called "degenerate process") or different energies (called "nondegenerate process"). An intermediate virtual state is created from the interaction between the electromagnetic field and matter. Two-photon absorption was theoretically predicted by Maria Göppert-Mayer in 1931 during her doctoral thesis (Göppert-Mayer,1931). Only after thirty years, the experimental observation of the phenomenon became possible, thanks to the invention of the LASER. The first experimental verification of 2PA was confirmed by Kaiser et al, who detected the fluorescence signal emitted by a europium-doped crystal (Kaiser and Garrett,1961), whose electrons were excited via a two-photon transition. Furthermore, this effect was experimentally proved by Isaac Abella (Abella,1962) in 1962, using Cesium vapor as the two-photon absorbing material. Nowadays, two-photon absorption processes have been extensively studied because of the enormous number of technological applications, such as, multiphoton optical limiters, multiphoton fluorescence spectroscopy, as well as multiphoton polymerization. The importance of multi-photon absorption studies is not only restricted to technological applications, but also lies in basic research, which enables to characterize and gather information about different materials. In this field, the quantification of such effect can provide insights on the molecular geometry and electronic charge distribution of an excited molecule. Additionally, two-photon absorption measurements provide properties (symmetry and energy) of some electronic states that can not be seen

To promote an excitation by absorption of two photons, the sum of the energies of each isolated photon need to match the energy of the electronic transition, as illustrated in Fig. 1,

et al., 2007, Lee, et al., 2008, Marder, et al., 2007, Sun and Kawata, 2004).

istics), desired for technological applications.

using traditional spectroscopic experiments.

**2. Two-photon absorption: Fundamentals**

**Figure 1.** Representation of a degenerated and nondegenarated two-photon transition between *n* and *n* 1 electronic states of an atom or molecule. The dotted lines represent a virtual state which intermediate the two-photon absorption process.

Due to the low probability of the 2PA to occur (compared to conventional one-photon absorption (1PA)), a high number of photons per unit of time and per unit of area is required to reach the atoms or molecules. As the effect exhibits a quadratic dependence on the number of photons, 2PA can be achieved by using ultrashort laser pulses. Once the strength of the two-photon absorption effect is several orders of magnitude smaller than the one-photon one, it is very important that the material presents no linear absorption at the laser wavelength employed.

In order to better understand how the two-photon effect appears in the light-matter interaction, one can imagine that when the electromagnetic field, which interacts with the matter, has a magnitude in the order of the interatomic field, it is able to modify the charge distribution in the material in a anharmonic way. As a consequence, the polarization of the material will be described by nonlinear terms, which are dependent on the incident electromagnetic field. Depending on the phase between the generated polarization and the applied electric field, the results can be either the Kerr effect (Abella,1962) or two-photon absorption. When two-photon absorption is analyzed from the nonlinear optics point of view, one can see that it is related to the imaginary part of the third-order nonlinear susceptibility*, (3)*, (Boyd,1992, Shen,1984), as shown by Eq. 1 (Sheik-Bahae,1990):

$$\operatorname{Im} \, \mathcal{X}^{(3)} = \mathcal{X}\_l^{(3)} = \frac{n\_0^2 \varepsilon\_0 c^2}{\alpha} \beta \tag{1}$$

in which *c* is the speed of light, *<sup>0</sup>* is the vacuum permittivity, is the photon frequency, *n0* is the refractive index of the material, and β is the two-photon absorption coefficient. Consequently, if the two-photon absorption process is significant, the total light absorption () of the material is described by:

$$
\alpha(I) = \alpha\_0 + \beta I\_0 \tag{2}
$$

where, *<sup>0</sup>* is the linear absorption, and *I0* is the light intensity reaching the sample. Equation 2 shows that the two-photon effect becomes more pronounced when the light intensity is considerable high. By using Eq. 2, the attenuation of light intensity as it propagates in the material (represented in Fig. 2), is described by:

$$\frac{dI}{dz} = -\left(\alpha\_0 I\_0 + \beta I\_0^2\right) \tag{3}$$

with *I* being the light intensity and *z* the optical path.

Considering that the material presents negligible linear absorption, the first term on the right side of Eq. 3 can be eliminated. Consequentially, the attenuation of the light will only depend on the two-photon absorption coefficient.

**Figure 2.** Representation of light attenuation when a materials of thickness L presents a two-photon absorption process.

After some mathematic manipulation, the solution for the attenuation of the light (*I*) due to the two-photon absorption process can be written as:

$$I = \frac{I\_0}{1 + \beta L I\_0} \tag{4}$$

By Eq. 4, one can see a decrease of the light intensity as the two-photon absorption coefficient increases. This coefficient carries information about the probability of a molecule or atom to absorb two photons simultaneously. In other words, the two-photon absorption coefficient is directly associated with the two-photon absorption cross-section (), and is described by Eq. 5:

336 Polymerization

where,

absorption process.

susceptibility*,*

in which *c* is the speed of light,

() of the material is described by:

material (represented in Fig. 2), is described by:

with *I* being the light intensity and *z* the optical path.

depend on the two-photon absorption coefficient.

the two-photon absorption process can be written as:

absorption. When two-photon absorption is analyzed from the nonlinear optics point of view, one can see that it is related to the imaginary part of the third-order nonlinear

(3) (3) 0 0 Im *<sup>I</sup>*

*<sup>0</sup>* is the vacuum permittivity,

the refractive index of the material, and β is the two-photon absorption coefficient. Consequently, if the two-photon absorption process is significant, the total light absorption

> ( )*I I*

shows that the two-photon effect becomes more pronounced when the light intensity is considerable high. By using Eq. 2, the attenuation of light intensity as it propagates in the

*dI I I*

Considering that the material presents negligible linear absorption, the first term on the right side of Eq. 3 can be eliminated. Consequentially, the attenuation of the light will only

*I0 <sup>I</sup>*

*(I) I*

**Figure 2.** Representation of light attenuation when a materials of thickness L presents a two-photon

After some mathematic manipulation, the solution for the attenuation of the light (*I*) due to

0 <sup>0</sup> 1 *<sup>I</sup> <sup>I</sup>*

*LI* (4)

*L*

0 0

*<sup>0</sup>* is the linear absorption, and *I0* is the light intensity reaching the sample. Equation 2

 <sup>2</sup> 00 0

*dz*

*(3)*, (Boyd,1992, Shen,1984), as shown by Eq. 1 (Sheik-Bahae,1990):

2 2

(1)

is the photon frequency, *n0* is

(2)

(3)

*n c*

$$
\beta = \frac{\delta N}{h\nu} \tag{5}
$$

where, *h* is the Planck constant, and *N* is the number of absorbers per unit of volume. The molecular two-photon absorption cross-section, δ, is usually given in units of Göppert-Mayer (GM), in which, 1 GM is equivalent to 1×10-50 cm4.s.photon-1. It is important to mention that similar to the one-photon absorption spectrum, a molecule or atom can present a two-photon absorption spectrum as well. Nowadays, in order to quantify and understand the two-photon absorption cross-sections spectra, monochromatic light sources have been employed. This is normally achieved when optical parametric amplifiers devices are used. In addition, supercontinuum light sources have also been employed to observe the magnitude of this effect (Balu, et al.,2004, De Boni, et al.,2004).

In terms of quantum mechanics analysis, two-photon absorption presents selection rules distinct from one-photon absorption ones. For example, if the molecule is centrosymmetric, one and two-photon transitions for a certain energy level are not allowed at the same time. In other words, one-photon allowed state is not a two-photon allowed one and vice-versa. This arises as a consequence from the angular momentum, which needs to be conserved after the electronic transition. Since photons present spin 1, one-photon absorption requires that the states involved in the transition differ in its orbital wavefunctions by an angular momentum of 1. In the same way, for a two-photon transition, the energy levels involved require a change of 2. This particular characteristic in the quantum mechanics selection rules turns the linear spectroscopy unable to indentify two-photon allowed states. In the opposite way, nonlinear spectroscopy, such as two-photon absorption measurements are no able to excite one-photon transitions. On the other hand, when a molecule is not centrosymmetric, the selection rules can be relaxed; enabling the existence of final states reached both by one- or two-photon absorption.

### **3. Two-photon absorption polymerization: Principles and applications**

Photopolymerization can be defined as a chemical reaction that turns molecules of low molecular weight (monomers) into macromolecules consisting of repeating units, by using light as the reaction trigger. For one-photon absorption polymerization, the radical species are formed by using a conventional light source (e.g. UV lamp). However, in a two-photon absorption polymerization, electronic transition occurs by simultaneous absorption of two photons, so that radical formation and subsequent polymerization occurs only in the vicinity of the focused light source (laser beam). This results in a small solidified volume (voxel) around the focal spot. By either scanning the laser beam through the resin volume or by

moving the sample in three perpendicular directions (X, Y and Z), arbitrary tridimensional microstructures can be created.

Also, by changing the monomer or the mixture of monomers used in the resin, one can change some characteristics of the final polymer. In this way, it is possible to tailor some properties of the produced microstructure. For example, polymer hardness and the amount of shrinkage the material undergoes due to polymerization can be controlled. Even more, by doping the base resin with different compounds, it is possible to change physical, chemical, biological and optical properties of the final microstructure.

two-photon polymerization microfabrication has a set of unique advantages over conventional microfabrication processes. Because several pulsed laser systems operate in the near infra-red region, where most curable monomers and polymers are transparent, and because two-photon absorption occurs only in defined spatial regions (where light intensity is high enough), it is possible to reach higher light penetration depths. In this manner, this process is able to polymerize, or solidify only a small region in the bulk of the liquid resin, without making any changes to its surface or the surrounding region.

The spatial confinement of the excitation, based on a two-photon process, is illustrated in Fig. 3, which depicts a comparison between one and two-photon induced fluorescence (Marder, et al.,2007) in a liquid sample contained in a silica fused cuvette. Note the fluorescence localization of the two-photon induced process, while one-photon excitation promotes fluorescence along the whole optical path.

**Figure 3.** Image of induced fluorescence for one and two-photon absorption process (1PA and 2PA). Note the fluorescence localization of two-photon induced processes. Reprinted with permission from (Marder, et al.,2007).Copyright [2007], Materials Research Society.

The resolution in microdevice fabrication technique is of prime importance. The nonlinear nature of two-photon polymerization allows for localized excitation and gives a fine resolution to this fabrication method. In fact, the two-photon absorption process pushes the fabrication resolution bellow the diffraction limit. All optical systems resolutions are limited by its optical instruments diffraction. The diffraction limit is the fundamental maximum resolution that any optical system can achieve. The minimum spot size that can be achieved is given by Abbe's expression (Fowles,1989):

338 Polymerization

microstructures can be created.

biological and optical properties of the final microstructure.

promotes fluorescence along the whole optical path.

(Marder, et al.,2007).Copyright [2007], Materials Research Society.

without making any changes to its surface or the surrounding region.

moving the sample in three perpendicular directions (X, Y and Z), arbitrary tridimensional

Also, by changing the monomer or the mixture of monomers used in the resin, one can change some characteristics of the final polymer. In this way, it is possible to tailor some properties of the produced microstructure. For example, polymer hardness and the amount of shrinkage the material undergoes due to polymerization can be controlled. Even more, by doping the base resin with different compounds, it is possible to change physical, chemical,

two-photon polymerization microfabrication has a set of unique advantages over conventional microfabrication processes. Because several pulsed laser systems operate in the near infra-red region, where most curable monomers and polymers are transparent, and because two-photon absorption occurs only in defined spatial regions (where light intensity is high enough), it is possible to reach higher light penetration depths. In this manner, this process is able to polymerize, or solidify only a small region in the bulk of the liquid resin,

The spatial confinement of the excitation, based on a two-photon process, is illustrated in Fig. 3, which depicts a comparison between one and two-photon induced fluorescence (Marder, et al.,2007) in a liquid sample contained in a silica fused cuvette. Note the fluorescence localization of the two-photon induced process, while one-photon excitation

**Figure 3.** Image of induced fluorescence for one and two-photon absorption process (1PA and 2PA). Note the fluorescence localization of two-photon induced processes. Reprinted with permission from

The resolution in microdevice fabrication technique is of prime importance. The nonlinear nature of two-photon polymerization allows for localized excitation and gives a fine resolution to this fabrication method. In fact, the two-photon absorption process pushes the fabri-

1PA

2PA

$$r = \frac{0.61\lambda}{NA} \tag{6}$$

where, *r* is the focal spot radius, *λ* is the light wavelength, and *NA* is the numerical aperture of the system. The numerical aperture is a number defined by Abbe as:

$$NA = n\sin(\theta)\tag{7}$$

where, *n* is the index of refraction in the focusing medium, and *θ* is the convergence angle of the beam. Figure 4 shows a schematic diagram for objectives with different *NA*.

**Figure 4.** Light focusing through microscope objective lens with different *NA*. W.D. stands for working distance.

The greater the numerical aperture, the smaller the working distance, i.e., the distance between the focal plane and the surface of the objective, according to Fig. 4. However, Abbe's equation works for plane waves only. Since most pulsed lasers have an approximately Gaussian beam profile, the minimum beam waist is given by:

$$
\omega v\_0 = \frac{\lambda}{\pi N \Lambda} \tag{8}
$$

where ω0 is the beam waist at the focal plane. The beam waist is defined as the distance from the center of the Gaussian beam up to the point where the amplitude of the electric field falls down to *1/e* of its maximum. However, Eq. 8 is still an approximated value; it works well for small *NA* optics (*NA*<<1). Generally, however, two-photon polymerization setups use high numerical aperture optics, such as microscope objective lenses. For high numerical aperture systems, such as immersion microscope objectives, a more accurate expression would be:

$$w\_0 = \frac{\lambda}{\pi NA} \sqrt{n^2 - NA^2} \tag{9}$$

Equation 9 shows that the smallest feature that one system can produce is limited by the microscope objective lens numerical aperture and by the wavelength. Nevertheless, once two-photon absorption rate is proportional to the square of the incident beam intensity and, therefore, depends on the squared Gaussian intensity profile, the beam presents a narrower waist than the one defined by Eq. 9.

Figure 5 shows the difference between a Gaussian and a squared Gaussian beam profile. Therefore, due to the nonlinear nature of this process, it is possible to achieve resolution below the diffraction limited spot size. This phenomenon reduces the beam waist by a factor of 2 . In addition to the square dependence, there is still another phenomenon that can yield even better resolution to two-photon polymerization process, which is the polymerization threshold. The polymerization threshold imposes a minimum power below which, even if the sample is irradiated, no polymerization takes place. This phenomenon usually occurs due to the presence of oxygen in the resin. Molecular oxygen inhibits the action of the photoinitiator radicals, preventing the polymerization reaction from occurring. Although this might be a problem for polymer film curing, it can be used as an advantage in the two-photon polymerization fabrication, in the sense that it can decrease even more the polymerized voxel size.

**Figure 5.** Gaussian and squared gaussian beam intensity profile. For comparison, it is also displayed (dotted line) the laser beam full width at half maximum (FWHM).

For a Gaussian profile, the central part of the beam is more intense than its borders. Therefore, even though the outer regions of the beam might not have enough power to start the reaction, the central part of the beam can overcome the threshold. It is important to realize that polymerization will also be contained in the Z direction. Equation 10 shows the intensity envelope of a Gaussian beam in cylindrical coordinates:

340 Polymerization

waist than the one defined by Eq. 9.

polymerized voxel size.

small *NA* optics (*NA*<<1). Generally, however, two-photon polymerization setups use high numerical aperture optics, such as microscope objective lenses. For high numerical aperture systems, such as immersion microscope objectives, a more accurate expression would be:

> 2 2 *w n NA* <sup>0</sup> *NA*

Equation 9 shows that the smallest feature that one system can produce is limited by the microscope objective lens numerical aperture and by the wavelength. Nevertheless, once two-photon absorption rate is proportional to the square of the incident beam intensity and, therefore, depends on the squared Gaussian intensity profile, the beam presents a narrower

Figure 5 shows the difference between a Gaussian and a squared Gaussian beam profile. Therefore, due to the nonlinear nature of this process, it is possible to achieve resolution below the diffraction limited spot size. This phenomenon reduces the beam waist by a factor of 2 . In addition to the square dependence, there is still another phenomenon that can yield even better resolution to two-photon polymerization process, which is the polymerization threshold. The polymerization threshold imposes a minimum power below which, even if the sample is irradiated, no polymerization takes place. This phenomenon usually occurs due to the presence of oxygen in the resin. Molecular oxygen inhibits the action of the photoinitiator radicals, preventing the polymerization reaction from occurring. Although this might be a problem for polymer film curing, it can be used as an advantage in the two-photon polymerization fabrication, in the sense that it can decrease even more the

**Figure 5.** Gaussian and squared gaussian beam intensity profile. For comparison, it is also displayed

(dotted line) the laser beam full width at half maximum (FWHM).

(9)

$$I(r,z) = I\_0 \frac{w\_0^2}{w(z)^2} e^{-2\left(\frac{r}{w(z)}\right)^2} \tag{10}$$

If we consider the laser beam waist at the focal plane and its intensity profile, only the central part of it will effectively solidify the resin. The beam waist (radius) at the focal plane is ω0 and it grows with *Z* (distance from the focal plane) as:

$$w(z) = w\_0 \sqrt{1 + \left(\frac{2\mathcal{L}}{\pi m w\_0^2}\right)} \equiv \sqrt{\frac{\mathcal{A}}{\pi m} \left(z\_R + \frac{z^2}{z\_R^2}\right)}\tag{11}$$

where *λ* is the wavelength, *n* is the index of refraction and *zR* is the Rayleigh length, defined as the distance from the focal plane in which the beam waist increases by a factor of 2 . The Rayleigh length can be written as:

$$z\_R = \frac{\pi m w\_0}{\lambda} \tag{12}$$

If we set *I(r,z) = Ith* in Eq. 5, (*th* stands for threshold) we can obtain the diameter (*D*) and the length (*L*) of the focal region where the intensity exceeds the threshold value (Juodkazis, et al.,2005), according to:

$$D(r) = w\_0 \sqrt{2 \ln \left( \frac{I(r)}{I\_{th}} \right)} \tag{13}$$

$$L(z) = 2z\_R \sqrt{\left(\frac{I(z)}{I\_{th}}\right) - 1} \tag{14}$$

As the two-photon absorption process depends on the square of the intensity and therefore on the focal volume in which the polymerization reaction in fact occurs, the diameter (*D*) and length (*L*) will be smaller. The dimensions of the polymerization voxel for two-photon polymerization should be written as (Juodkazis, et al, 2005):

$$D(r) = w\_0 \sqrt{\ln\left(\frac{I(r)}{I\_{th}}\right)}\tag{15}$$

$$L(z) = 2z\_R \sqrt{\left(\frac{I(z)}{I\_{th}}\right)^{1/2} - 1} \tag{16}$$

By controlling laser power in order to have it near the polymerization threshold value, it is possible to create structures with resolution well below the diffraction limit. Figure 6 illustrates how polymerization threshold can yield further improvements in fabrication resolution regardless of the optical system. These are some of the characteristics that made twophoton polymerization microfabrication technique so appealing for many research groups in the last decade.

**Figure 6.** Threshold power relative to various gaussian profiles. The more the peak power gets closer to the threshold power the smaller is the voxel size. Polymerization is not observed if peak power is below the threshold power.

The first two-photon polymerized microstructures were reported by Maruo et al (Maruo, et al.,1997) and date back to 1997. In that work, their intention was to demonstrate the capabilities of this new fabrication technique. No other process offers both, complex tridimensional fabrication and resolution below the diffraction limit. In order to illustrate such features, Kawata et al (Kawata, et al.,2001) used a computer model and created a micro sculpture of a bull with subdiffraction-limit resolution. In addition, to show that the microstructures retained good structural and physical properties, they fabricated a micro spring attached to one cube and demonstrated its oscillation. Figure 7 (a) and (b) shows both structures.

in the last decade.

the threshold power.

7 (a) and (b) shows both structures.

1/2

(16)

( ) () 2 <sup>1</sup> *<sup>R</sup> th*

By controlling laser power in order to have it near the polymerization threshold value, it is possible to create structures with resolution well below the diffraction limit. Figure 6 illustrates how polymerization threshold can yield further improvements in fabrication resolution regardless of the optical system. These are some of the characteristics that made twophoton polymerization microfabrication technique so appealing for many research groups

**Figure 6.** Threshold power relative to various gaussian profiles. The more the peak power gets closer to the threshold power the smaller is the voxel size. Polymerization is not observed if peak power is below

The first two-photon polymerized microstructures were reported by Maruo et al (Maruo, et al.,1997) and date back to 1997. In that work, their intention was to demonstrate the capabilities of this new fabrication technique. No other process offers both, complex tridimensional fabrication and resolution below the diffraction limit. In order to illustrate such features, Kawata et al (Kawata, et al.,2001) used a computer model and created a micro sculpture of a bull with subdiffraction-limit resolution. In addition, to show that the microstructures retained good structural and physical properties, they fabricated a micro spring attached to one cube and demonstrated its oscillation. Figure

*I*

*I z Lz z*

**Figure 7.** Micro sculpture of (a) a bull and (b) oscillating spring system. Scale bar is 2µm. Reprinted with permission from (Kawata, et al.,2001). Copyright [2001], Nature Publishing Group.

#### **4. Materials employed and experimental setup**

The polymeric resins employed for two-photon absorption polymerization vary widely. For instance, Maruo et al reported the use of urethane acrylate monomer (Maruo, et al.,1997), Baldacchini et al (Baldacchini, et al.,2004) and Haske et al (Haske, et al.,2007) used a mixture of two-acrylic resin, while Chichkov's group employed Inorganic-organic hybrid polymers (Schlie, et al.,2007, Serbin, et al.,2004). Our group has used the same resin formulation proposed by Baldacchini et al (Baldacchini, et al.,2004), whose formulation consists of a mixture of two tri-acrylate monomers (see Fig. 8): (a) tris(2-hydroxyethyl)isocyanurate triacrylate and (b) ethoxylated(6) trimethylolpropane triacrylate. The first one increases the microstructure hardness, while the second one reduces shrinkage upon polymerization. Ethyl-2,4,6 trimethylbenzoylphenylphosphinate, commercially known as Lucirin TPO-L, has been used as the photoinitiator, due to its favorable two-photon absorption properties (Baldacchini, et al.,2004, Mendonca, et al.,2008). The chemical structures of the two acrylic resins and the photoinitiator are displayed in Fig. 8 (a), (b) and (c) respectively.

**Figure 8.** Chemical structures of materials used in the two-photon polymerization. (a) tris(2 hydroxyethyl)isocyanurate triacrylate and (b) ethoxylated(6) trimethylolpropane triacrylate and (c) Ethyl-2,4,6-trimethylbenzoylphenylphosphinate (photoinitiator).

The experimental setup used in two-photon polymerization typically employs femtosecond lasers and pulse energies of nanojoules. Although the average energy is low, the pulse peak intensity is high enough to promote two-photon absorption polymerization. In our specific case, we use a Ti:sapphire laser oscillator system, which delivers pulses of 100 fs centered at 800 nm. The laser beam is focused onto the resin to be polymerized by using a 0.65-NA microscope objective. The average power used depends on the nature of the materials employed (such as type of polymeric resin, photoinitiator, dopants etc), once it must be high enough to overcome polymerization threshold, but low enough to avoid material degradation. We typically employ pulse energies in the order of 0.1 nJ, measured before the objective. The sample preparation consists in placing a drop of the resin on a substrate (glass slide) previously cleaned. The resin is retained between spacers and enclosed by a cover slip, as illustrated in Fig. 9.

The sample is then positioned and moved in the axial *z* direction using a motorized stage, while the laser beam scans the sample across the *xy* plane, by using a pair of galvano mirrors. In addition, a CCD camera is coupled to the experimental setup to monitor two-photon absorption polymerization in real time. An illustration of the two-photon polymerization experimental setup is displayed in Fig. 10.

After the desired microstructure is fabricated, the sample is immersed in ethanol to wash away the unsolidified resin, leaving behind solely the desired microstructure on the glass substrate, according to the scheme sequence depicted in Fig. 11.

**Figure 10.** Scheme of the experimental setup employed for the two-photon absorption polymerization. M, BS, SM-x and SM-y stands for mirror, beam splitter, scanning mirror in x and y direction, respectively.

**Figure 11.** Scheme of the sample washing with ethanol after microfabrication to remove the unpolymerized resin.

#### **5. Doped microstructures and applications**

344 Polymerization

slip, as illustrated in Fig. 9.

**cover slip**

between a glass slide and a cover slip, separated by spacers.

substrate, according to the scheme sequence depicted in Fig. 11.

experimental setup is displayed in Fig. 10.

**spacer**

**resin**

The experimental setup used in two-photon polymerization typically employs femtosecond lasers and pulse energies of nanojoules. Although the average energy is low, the pulse peak intensity is high enough to promote two-photon absorption polymerization. In our specific case, we use a Ti:sapphire laser oscillator system, which delivers pulses of 100 fs centered at 800 nm. The laser beam is focused onto the resin to be polymerized by using a 0.65-NA microscope objective. The average power used depends on the nature of the materials employed (such as type of polymeric resin, photoinitiator, dopants etc), once it must be high enough to overcome polymerization threshold, but low enough to avoid material degradation. We typically employ pulse energies in the order of 0.1 nJ, measured before the objective. The sample preparation consists in placing a drop of the resin on a substrate (glass slide) previously cleaned. The resin is retained between spacers and enclosed by a cover

**Laser beam**

**Figure 9.** Scheme of sample preparation, where the resin to be polymerized by the laser beam is placed

The sample is then positioned and moved in the axial *z* direction using a motorized stage, while the laser beam scans the sample across the *xy* plane, by using a pair of galvano mirrors. In addition, a CCD camera is coupled to the experimental setup to monitor two-photon absorption polymerization in real time. An illustration of the two-photon polymerization

After the desired microstructure is fabricated, the sample is immersed in ethanol to wash away the unsolidified resin, leaving behind solely the desired microstructure on the glass

**objective**

**glass slide**

Most of the initial studies reporting microfabrication of structures via two-photon absorption polymerization dealt with undoped microstructures. Although in this case the structures still present high definition and functionality, the final structures are of limited application, once the properties cannot be probed or altered by external sources. In the last few years, however, several research groups have attempted to dope the resin formulations with a large variety of advanced materials, aiming at giving new functionalities to microstructures, such as chemical, biological or optical properties. Next we will present some results that diverse research groups have obtained by using methodologies to

incorporate distinct dopants in the resin formulations, aiming at obtaining two-photon polymerized microstructures with unique properties for technological applications.

#### - *Incorporation of conjugated polymers*

Conjugated polymers are materials of great interest to be used in technological devices, once the electron delocalization along the polymer backbone can render them conductivity and nonlinear optical properties. For instance, in a methodology proposed by our group (Mendonca, et al.,2009), poly (2-methoxy-5-2-ethylhexyloxy-1,4-phenylenevinylene) (MEH-PPV), widely known for its conductivity (Yu, et al.,1995), electroluminescence (Nguyen, et al.,2000, Parker,1994), and nonlinear optical properties (Chung, et al.,2002, Correa, et al.,2007, De Boni, et al.,2004, Oliveira, et al.,2006) has been incorporated into the basic resin in a guesthost strategy. In the proposed methodology, the host consists of the two triacrylate monomers and the photoinitiator displayed in Fig. 8. To this solution, it is added up to 1.5% of MEH-PPV, which was then mixed up for 2 hours. Ethanol was eliminated by evaporation at room temperature for 24 h, yielding a viscous liquid. From this liquid, we extracted some drops and place on the glass substrate, as illustrated in Fig. 9.

Two-photon absorption polymerization was employed to fabricate waveguides containing MEH-PPV, using as substrates mesoporous silica films, which have a low refractive index (*n*  = 1.185) throughout the visible spectrum and, therefore, minimize waveguiding losses (Konjhodzic, et al.,2005). Figure 12 (a) shows a fluorescence microscopy top view image of 100 m-long waveguide fabricated on a mesoporous silica substrate, illuminated in its central region by a CW laser beam at 532 nm. The MEH-PPV fluorescence is guided through the microstructure and scatters at both microstructure ends. When the experiment was carried out in waveguides fabricated on top of a conventional glass slide, no waveguiding was observed due to light coupling to the substrate. Figure 12 (b) displays a side view scheme of the microstructure excitation and the corresponding waveguiding. Such waveguide is promising to be used in photonics applications such as microLEDs.

**Figure 12.** (a) Fluorescence optical microscopy (top view image) of a waveguide containing MEH-PPV fabricated by two-photon polymerization on mesoporous silica substrate. Light is guided through the microstructure ends when it is excited at its center. In (b) is displayed a schematic side view of the structure. Reprinted with permission from (Mendonca, et al.,2009).Copyright [2009], American Institute of Physics.

#### - *Incorporation of azoaromatic dyes*

346 Polymerization

drops and place on the glass substrate, as illustrated in Fig. 9.

ising to be used in photonics applications such as microLEDs.

(a)

(b)

incorporate distinct dopants in the resin formulations, aiming at obtaining two-photon

Conjugated polymers are materials of great interest to be used in technological devices, once the electron delocalization along the polymer backbone can render them conductivity and nonlinear optical properties. For instance, in a methodology proposed by our group (Mendonca, et al.,2009), poly (2-methoxy-5-2-ethylhexyloxy-1,4-phenylenevinylene) (MEH-PPV), widely known for its conductivity (Yu, et al.,1995), electroluminescence (Nguyen, et al.,2000, Parker,1994), and nonlinear optical properties (Chung, et al.,2002, Correa, et al.,2007, De Boni, et al.,2004, Oliveira, et al.,2006) has been incorporated into the basic resin in a guesthost strategy. In the proposed methodology, the host consists of the two triacrylate monomers and the photoinitiator displayed in Fig. 8. To this solution, it is added up to 1.5% of MEH-PPV, which was then mixed up for 2 hours. Ethanol was eliminated by evaporation at room temperature for 24 h, yielding a viscous liquid. From this liquid, we extracted some

Two-photon absorption polymerization was employed to fabricate waveguides containing MEH-PPV, using as substrates mesoporous silica films, which have a low refractive index (*n*  = 1.185) throughout the visible spectrum and, therefore, minimize waveguiding losses (Konjhodzic, et al.,2005). Figure 12 (a) shows a fluorescence microscopy top view image of 100 m-long waveguide fabricated on a mesoporous silica substrate, illuminated in its central region by a CW laser beam at 532 nm. The MEH-PPV fluorescence is guided through the microstructure and scatters at both microstructure ends. When the experiment was carried out in waveguides fabricated on top of a conventional glass slide, no waveguiding was observed due to light coupling to the substrate. Figure 12 (b) displays a side view scheme of the microstructure excitation and the corresponding waveguiding. Such waveguide is prom-

**Figure 12.** (a) Fluorescence optical microscopy (top view image) of a waveguide containing MEH-PPV fabricated by two-photon polymerization on mesoporous silica substrate. Light is guided through the microstructure ends when it is excited at its center. In (b) is displayed a schematic side view of the structure. Reprinted with permission from (Mendonca, et al.,2009).Copyright [2009], American Institute of Physics.

polymerized microstructures with unique properties for technological applications.

Farsari et al (Farsari, et al.,2008) reported the fabrication of photonic crystals via two-photon absorption polymerization, by incorporating into the basic resin a nonlinear optically active chromophore, based on reacting Disperse Red 1 (DR1) with (3-isocyanopropyl) triethoxysilane. The structures reported by them had a bright red color, typical of Disperse Red 1 and presented stop-gaps at specific frequencies, close to the near infrared region, typical of photonic crystals.

Birefringence in microstructures have been explored by Mendonça et al (Mendonca, et al.,2007), where they have fabricated, via two-photon absorption polymerization, microstructures incorporating the azodye DR13, as the one displayed in the SEM image in Fig. 13 (a). Birefringence could be optically induced and erased in these microstructures due to a reversible trans-cis photoisomerization and a subsequent molecular orientation of the azo group. After exposure to laser, the microstructure exhibited residual birefringence because the chromophore molecules become oriented in the direction perpendicular to the laser polarization. Figure 13 (b) and (c) show the corresponding transmission microscopy images, where the microstructure is visible when the angle between the sample axis, dened by the Ar+ ion laser exposure and the polarizers, is an odd multiple of 45°. This birefringence can be completely erased with circularly polarized light or by heating the sample close to the polymer glass transition temperature.

**Figure 13.** (a) Scanning electron micrograph of a cubic microstructure containing DR13. Polarization microscope images of a microstructure (like the one shown in (a)) at two angles between the polarizer and sample axis, displaying the birefringence behavior owing to DR13 incorporation. Reprinted with permission from (Mendonca, et al.,2007). Copyright [2007], American Institute of Physics.

#### - *Incorporation of magnetic nanoparticles*

Desinging and fabricating microdevices by two-photon absorption polymerization that would be easily manipulated are import to understand the mechanics on the micro/nanomachine motion. Wang et al, (Wang, et al.,2009) proposed an interesting methodology where the basic resin was doped with surface-modied Fe3O4 nanoparticles, and yielded structures with magnetic properties, whose movement could be externally controlled by magnets. Figure 14 displays a magnetic microspring fabricated by two-photon absorption polymerization, where one end of the spring remains attached to an anchor, which in turn, is attached to the substrate, and the other end is attached to a freely moving sphere. By applying a magnetic field, the objected was moved in the direction of higher magnetic eld strength. Therefore, by changing the ferromagnet position, distinct motions could be induced, as displayed in Fig. 14 (a) and (b), which shows the microspring in its non-alongated state (a) and in its alongated state (b).

**Figure 14.** Photograph of a magnetic micro-spring fabricated via two-photon absorption polymerization, by using a resin doped with magnetic nanoparticles. (a) Microspring in its natural non-alongated state and (b) in its alongated state. Reprinted with permission from (Wang, et al.,2009). Copyright [2009], Optical Society of America.

#### - *Incorporation of bio-related materials*

Two-photon polymerized structures incorporating biomaterials have also been subject of investigation by a few research groups. For instance, chitosan-containg microstructures fabricated by two-photon polymerization have been fabricated by Correa et al (Correa, et al.,2009). Chitosan is a biodegradable and biocompatible polymer, with applications in blood coagulation, soft tissue and bone regeneration, which could render to the microstructures appealing properties for medical applications. In that work, it was demonstrated, by Raman microscopy, that chitosan does not react chemically with the matrix resin, retaining its characteristics after the fabrication process, which is suitable for biomedical applications.

A methodology employing two-photon absorportion polymerization to fabricate antibactericide microneddles was proposed by Gittard et al (Gittard, et al.,2010). Nowadays, the use of microneedles for drug delivery is still limited because of the risk of infection associated with contamination of the micronnedles. Such problem may be overcome by imparting these devices with antimicrobial properties. Gittard et al (Gittard, et al.,2010) used a photopolymerizationmicromolding technique to fabricate microneedle arrays containing Polyethylene Glycol-Gentamicin Sulfate, which inhibited growth of *Staphylococcus aureus* bacteria.

348 Polymerization

state (a) and in its alongated state (b).

[2009], Optical Society of America.

cro/nanomachine motion. Wang et al, (Wang, et al.,2009) proposed an interesting methodology where the basic resin was doped with surface-modied Fe3O4 nanoparticles, and yielded structures with magnetic properties, whose movement could be externally controlled by magnets. Figure 14 displays a magnetic microspring fabricated by two-photon absorption polymerization, where one end of the spring remains attached to an anchor, which in turn, is attached to the substrate, and the other end is attached to a freely moving sphere. By applying a magnetic field, the objected was moved in the direction of higher magnetic eld strength. Therefore, by changing the ferromagnet position, distinct motions could be induced, as displayed in Fig. 14 (a) and (b), which shows the microspring in its non-alongated

**Figure 14.** Photograph of a magnetic micro-spring fabricated via two-photon absorption polymerization, by using a resin doped with magnetic nanoparticles. (a) Microspring in its natural non-alongated state and (b) in its alongated state. Reprinted with permission from (Wang, et al.,2009). Copyright

Two-photon polymerized structures incorporating biomaterials have also been subject of investigation by a few research groups. For instance, chitosan-containg microstructures fabricated by two-photon polymerization have been fabricated by Correa et al (Correa, et al.,2009). Chitosan is a biodegradable and biocompatible polymer, with applications in blood coagulation, soft tissue and bone regeneration, which could render to the microstructures appealing properties for medical applications. In that work, it was demonstrated, by Raman microscopy, that chitosan does not react chemically with the matrix resin, retaining its characteristics after

A methodology employing two-photon absorportion polymerization to fabricate antibactericide microneddles was proposed by Gittard et al (Gittard, et al.,2010). Nowadays, the use of microneedles for drug delivery is still limited because of the risk of infection associated with contamination of the micronnedles. Such problem may be overcome by imparting these devices

the fabrication process, which is suitable for biomedical applications.

(a) (b)

Claeyssens et al (Claeyssens, et al.,2009) proposed using two-photon polymerization to fabricate microstructures by using the biodegradable triblock copolymer poly(εcaprolactone-co-trimethylenecarbonate)-b-poly(ethyleneglycol)-b-poly(ε-caprolactone-cotrimethylenecarbonate). Samples of the structures fabricated are displayed in the SEM images in Fig. 15, which present good resolution. Besides, the initial cytotoxicity tests employed showed that the material does not affect cell proliferation, which demonstrates the capability of using two-photon polymerization to fabricate biodegradable scaffolds for tissue engineering.

**Figure 15.** Scanning electron micrographs of structures fabricated with biodegradable triblock copolymer poly(ε-caprolactone-co-trimethylenecarbonate)-b-poly(ethyleneglycol)-b-poly(ε-caprolactone-cotrimethylenecarbonate). Reprinted with permission from (Claeyssens, et al.,2009). Copyright [2009], American Chemical Society.

Farsari et al (Drakakis, et al.,2006, Farsari, et al.,2010) have functionalized the surface of tridimensional structures made with a biocompatible organic-inorganic hybrid matrix, and subsequently immobilized photosensitive biotin on it. The integrity of biotin was confirmed by binding and detecting the presence and distribution of avidin via uorescence microscopy. Experiments with peptides have also been carried out by the same group. Such methodology seems promising for applications in scaffolds for cell growth and tissue engineering.

#### - *Incorporation of other organic and inorganic materials*

Research groups have used a considerable variety of doping materials and methodologies for incorporating such materials into the base resins, in order to render, for instance, optical properties to the two-photon polymerized microstructures. Sun et al (Sun, et al.,2007) proposed a methodology to produce microstructures containing embedded CdS nanoparticles. The materials were prepared by using Cd acrylate derivatives as the base resin, and after the microfabrication process, the structures were treated with H2S gas for about 48 h to form CdS-polymer nanocomposites. By using this methodology, they were able to fabricate photonic crystals and structures presenting a high fluorescence signal.

Jia et al (Jia, et al.,2010) fabricated microstructures using a organic–inorganic hybrid photosensitive polymer containing dispersed PbS quantum dots. Such material exhibits high third-order nonlinearity, and was used to fabricate woodpile-shape photonic crystals with stop gaps in the telecommunication wavelength region. The authors also prospected these nanocomposite materials for applications in active micro/nanodevices, such as ultrafast switching, signal regeneration, and high speed demultiplexing systems.

Shukla et al (Shukla, et al.,2010) demonstrated a new method for producing subwavelength plasmonic and conductive patterned structures within a polymeric host. The method involves both, two-photon-initiated photoreduction of a gold precursor and two-photon polymerization using a negative photoresist. By this methodology, they several types of structures, including optically active planar chiral structures and plasmonic nanostructures, which have potential use for the development of metamaterials. Example of such structures are displayed in Fig. 16 (a) and (b) (Shukla, et al.,2010), which shows confocal microscopy plasmonic donuts fabricated by the mentioned method.

**Figure 16.** (a,b). Confocal uorescence image of plasmonic donuts structures fabricated by a combination of two-photon-initiated photoreduction of a gold precursor and two-photon absorption polymerization. Reprinted with permission from (Shukla, et al.,2010). Copyright [2010], American Chemical Society.

Microstructures containing rodhamine have been fabricated by two-photon absorption polymerization by a few research groups (Correa, et al.,2011, Zukauskas, et al.,2010). Rhodamine is a remarkable material to dope resins owing to its chemical stability, strong luminescence and nonlinear optical properties. Correa et al. (Correa, et al.,2011) fabricated two-photon polymerized structures containing rodhamine and excited them with silica wires with diameters of approximately 1 m, , which are produced through the fiber tapering technique (flame-heated fiber drawing) (Tong, et al.,2003). Figure 17 (a) shows a schematic view of the excitation of a two-photon polymerized microstructure with silica tapered fiber. Figure 17 (b) and (c) show optical micrographs of (a) frontal and (b) lateral excitation at 514 nm of microstructures containing rhodamine by tapered silica fibers (Correa, et al.,2011, Correa, et al.,2012).

Society.

Jia et al (Jia, et al.,2010) fabricated microstructures using a organic–inorganic hybrid photosensitive polymer containing dispersed PbS quantum dots. Such material exhibits high third-order nonlinearity, and was used to fabricate woodpile-shape photonic crystals with stop gaps in the telecommunication wavelength region. The authors also prospected these nanocomposite materials for applications in active micro/nanodevices, such as ultrafast

Shukla et al (Shukla, et al.,2010) demonstrated a new method for producing subwavelength plasmonic and conductive patterned structures within a polymeric host. The method involves both, two-photon-initiated photoreduction of a gold precursor and two-photon polymerization using a negative photoresist. By this methodology, they several types of structures, including optically active planar chiral structures and plasmonic nanostructures, which have potential use for the development of metamaterials. Example of such structures are displayed in Fig. 16 (a) and (b) (Shukla, et al.,2010), which shows confocal microscopy plasmonic donuts

**Figure 16.** (a,b). Confocal uorescence image of plasmonic donuts structures fabricated by a combination of two-photon-initiated photoreduction of a gold precursor and two-photon absorption polymerization. Reprinted with permission from (Shukla, et al.,2010). Copyright [2010], American Chemical

Microstructures containing rodhamine have been fabricated by two-photon absorption polymerization by a few research groups (Correa, et al.,2011, Zukauskas, et al.,2010). Rhodamine is a remarkable material to dope resins owing to its chemical stability, strong luminescence and nonlinear optical properties. Correa et al. (Correa, et al.,2011) fabricated two-photon polymerized structures containing rodhamine and excited them with silica wires with diameters of approximately 1 m, , which are produced through the fiber tapering technique (flame-heated fiber drawing) (Tong, et al.,2003). Figure 17 (a) shows a schematic view of the excitation of a two-photon polymerized microstructure with silica tapered fiber. Figure 17 (b) and (c) show optical micrographs of (a) frontal and (b) lateral excitation at 514 nm of microstructures containing rhodamine by tapered silica fibers

switching, signal regeneration, and high speed demultiplexing systems.

(a) (b)

fabricated by the mentioned method.

(Correa, et al.,2011, Correa, et al.,2012).

**Figure 17.** Excitation at 514 nm of a two-photon polymerized microstructure doped with rhodamine using silica tapered fiber. (a) Schematic view, (b) frontal excitation and (c) lateral excitation. (b) and (c) reprinted with permission from (Correa, et al.,2011). Copyright [2011], Springer Science+Business Media B.V.

To further advance the use of the structures' optical properties, integration between them was explored, aiming at exciting or collecting light by using silica tapered fibers. The integration between structures was carried out with silica tapered fibers similar to the one displayed in Fig. 17. Figure 18 shows a set of two-photon polymerized structures connect by such tapered optical fibers. This step represents a potential approach to integrate arrays of optical microstructures, aiming at the fabrication of micro-optical photonics circuitry.

**Figure 18.** Transmission optical micrograph of a set of two-photon polymerized structures connected by tapered optical fibers.

By incorporating two or more distinct dopants into the base resin, one can obtain twophoton polymerized microstructures that present, for instance, fluorescence at two or more distinct wavelengths. This approach would be useful, for instance, to produce multi-colored pixels presenting a RGB pattern. In Fig. 19, we show microscope images of undoped (gray) and rhodamine doped (purple) microstructures fabricated by two-photon polymerization in the same substrate.

**Figure 19.** Optical microcopy image of two-photon polymerized structures fabricated on the same substrate. The gray structure was fabricated only with the base resin (undoped), while the purple one (doped) contains rhodamine.

In addition to the based on the methodology of incorporating materials into the bulk of basic resin, another interesting approach is to functionalize the fabricated microstructure's surface. In this case, after functionalization, other materials can be adhered to the surface, giving to the microstructures unique properties. For instance, functionalization has been carried out by several groups to achieve metallization of structures' surface, with gold, silver, and copper (Farrer, et al.,2006, Rill, et al.,2008, Takeyasu, et al.,2008). Such metallic structures can be highly conductive, opening doors for applications in microelectrical circuitry.

## **6. Conclusions**

This chapter aimed at demonstrating fundamental and experimental aspects of fabrication of doped microstructures via two-photon absorption polymerization. Due to the efforts of many research groups in the last years, the use of this technique and its applications has spread out all over the world. By using this technique, tridimensional microstructures of complex shape, high definition and structural integrity can be fabricated to be used in microdevices. Distinct dopants incorporated into the basic resin formulation can yield to the microstructure features such as biocompatibility, luminescence, magnetic properties and waveguiding. Such microstructures with enhanced properties are highly desired for applications in optical devices, photonic crystals, micro-waveguides, microfluidics and scaffolds for tissue engineering, whose commercial applications in the market are expected to happen in the next years.

## **Author details**

Daniel S. Correa *Embrapa Instrumentação, São Carlos, SP, Brazil*

Leonardo De Boni, Adriano J. G. Otuka, Vinicius Tribuzi and Cleber R. Mendonça *Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, SP, Brazil*

## **Acknowledgement**

352 Polymerization

the same substrate.

(doped) contains rhodamine.

**6. Conclusions**

the next years.

**Author details**

*Embrapa Instrumentação, São Carlos, SP, Brazil*

Daniel S. Correa

and rhodamine doped (purple) microstructures fabricated by two-photon polymerization in

**Figure 19.** Optical microcopy image of two-photon polymerized structures fabricated on the same substrate. The gray structure was fabricated only with the base resin (undoped), while the purple one

conductive, opening doors for applications in microelectrical circuitry.

In addition to the based on the methodology of incorporating materials into the bulk of basic resin, another interesting approach is to functionalize the fabricated microstructure's surface. In this case, after functionalization, other materials can be adhered to the surface, giving to the microstructures unique properties. For instance, functionalization has been carried out by several groups to achieve metallization of structures' surface, with gold, silver, and copper (Farrer, et al.,2006, Rill, et al.,2008, Takeyasu, et al.,2008). Such metallic structures can be highly

This chapter aimed at demonstrating fundamental and experimental aspects of fabrication of doped microstructures via two-photon absorption polymerization. Due to the efforts of many research groups in the last years, the use of this technique and its applications has spread out all over the world. By using this technique, tridimensional microstructures of complex shape, high definition and structural integrity can be fabricated to be used in microdevices. Distinct dopants incorporated into the basic resin formulation can yield to the microstructure features such as biocompatibility, luminescence, magnetic properties and waveguiding. Such microstructures with enhanced properties are highly desired for applications in optical devices, photonic crystals, micro-waveguides, microfluidics and scaffolds for tissue engineering, whose commercial applications in the market are expected to happen in

Leonardo De Boni, Adriano J. G. Otuka, Vinicius Tribuzi and Cleber R. Mendonça *Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, SP, Brazil*

We thank financial support from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) , Conselho Nacional de Desenvolvimento Cientíco Tecnológico (CNPq), and Embrapa.

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## **A Survey of Equations of State for Polymers**

Yuri Guerrieri, Karen Valverde Pontes, Gloria Meyberg Nunes Costa and Marcelo Embiruçu

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48391

## **1. Introduction**

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The thermodynamics of polymeric systems play an important role in the polymer industry and are often a key factor in polymer production, processing and material development, especially for the design of advanced polymeric materials. Many polymeric products are produced with a solvent or diluent (or a mixture of them) and often with other low molecular weight compounds (plasticizers, among others). A problem which often arises is how to remove the low molecular weight constituent(s) from the final product (polymer). The solution to this problem involves, among other tasks, solving the vapor-liquid equilibrium (VLE) and/or the vapor-liquid-liquid equilibrium (VLLE) problem. Other applications of polymer thermodynamics directly involve the polymerization processes. For example, several processes such as the production of PET (polyethylene terephthalate) are carried out in two-phase (vapor-liquid) reactors. Phase equilibrium compostions of the reacting components will determine their phase concentrations and thus the outcome of the polymerization reaction. Another example is the case of LDPE (Low Density PolyEthylene) made in autoclave reactors where it may be desirable to perform the polymerization reaction nearby but outside the two-liquid phase region, but close to it, which makes accurate liquid-liquid equilibrium (LLE) information at high pressure essential. During PE (polyethylene) or PP (polypropylene) industrial processing, for example, deposition of the polymer on the reactor surface, heat exchangers and flash drums frequently occurs and this can cause clogging in pipelines. Modeling solid-liquid equilibrium (SLE) is a useful basis from which to gain a better understanding of these industrial polymer problems and thus to avoid their occurrence.

Analogous to the modeling of conventional phase equilibrium, there are two basic approaches available to describe phase equilibrium of polymer-solvent mixtures: activity coefficient models and equations of state (EOS). There are several drawbacks to the activity coefficient approach, for example: it is hard to define standard states, especially for

© 2012 Pontes et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Pontes et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

supercritical components; the parameters of the activity coefficient models are very temperature dependent, and critical phenomena are not predicted because different models are used for the vapor and liquid phases. Furthermore, other thermodynamic properties such as densities, enthalpies, entropies, among others, cannot usually be obtained from the same model because the excess Gibbs free energy is rarely known as a function of temperature and pressure.

EOS are powerful tools for investigating thermodynamic properties and phase behavior of pure fluids and their mixtures. There are many well-tested EOS available for fluid mixtures of conventional substances. For mixtures of polymers with solvents, on the other hand, problems arise due to the different characteristics of the components. To address these many polymer-specific EOS have been proposed, which focus on the polymer component(s) of the mixture. Efforts to represent conventional systems with these EOS have not always been very successful; indeed some of these models perform less successfully than traditional cubic EOS in this regard. This may be a handicap when these models are used for the VLE of the polymer-solvent mixtures. In such cases, little or no polymer is present in the vapor phase and the solvent compressibility plays an important role in the phase behavior. Consequently, there is a strong incentive to extend the conventional EOS developed for small molecules to polymers.

There are two basic issues in extending cubic EOS to apply to polymers and their use. The first issue is the description of the pure component EOS parameters for polymers. To obtain these parameters, various techniques have been suggested. The second issue in extending cubic EOS to apply to polymers is the selection of mixing rules (MR) for the EOS parameters. The classical mixing rules of van der Waals (vdW) have already been tested for polymer solvent mixtures, however, it has been observed that, in order to fit the experimental data, some unrealistic values are necessary for the binary interaction parameters (BIP).

The use of equations of state in phase equilibrium modeling instead of activity coefficient models is mainly a result of the recent development of a class of mixing rules that enable the use of liquid activity coefficient models in the EOS formalism. The implication of this change is far-reaching as an EOS offers a unified approach in thermodynamic property modeling. With this approach, the applicability of simple cubic EOS has been extended to complex systems, such as polymeric systems, if coupled with the appropriate activity coefficient model. Therefore, there is much interest in mixture EOS models capable of describing higher degrees of nonideality than that possible with the van der Waals one-fluid model and its modifications.

Future development of EOS for polymer mixtures is unclear and some contradictory statements can be found in the literature. Some authors indicate that cubic equations can be extended to correlate and predict VLE in polymer mixtures accurately. On the other hand, others state that, considering the complexity of this type of mixture, simplicity is not a necessary requirement for an EOS, as the calculation of parameters for the mixture components is more important. There is agreement, however, on the fact that future development of EOS for polymer mixtures must emphasize the study of mixing rules and that EOS input parameters should be related to the commonly measured properties of the polymers.

358 Polymerization

temperature and pressure.

small molecules to polymers.

parameters (BIP).

model and its modifications.

supercritical components; the parameters of the activity coefficient models are very temperature dependent, and critical phenomena are not predicted because different models are used for the vapor and liquid phases. Furthermore, other thermodynamic properties such as densities, enthalpies, entropies, among others, cannot usually be obtained from the same model because the excess Gibbs free energy is rarely known as a function of

EOS are powerful tools for investigating thermodynamic properties and phase behavior of pure fluids and their mixtures. There are many well-tested EOS available for fluid mixtures of conventional substances. For mixtures of polymers with solvents, on the other hand, problems arise due to the different characteristics of the components. To address these many polymer-specific EOS have been proposed, which focus on the polymer component(s) of the mixture. Efforts to represent conventional systems with these EOS have not always been very successful; indeed some of these models perform less successfully than traditional cubic EOS in this regard. This may be a handicap when these models are used for the VLE of the polymer-solvent mixtures. In such cases, little or no polymer is present in the vapor phase and the solvent compressibility plays an important role in the phase behavior. Consequently, there is a strong incentive to extend the conventional EOS developed for

There are two basic issues in extending cubic EOS to apply to polymers and their use. The first issue is the description of the pure component EOS parameters for polymers. To obtain these parameters, various techniques have been suggested. The second issue in extending cubic EOS to apply to polymers is the selection of mixing rules (MR) for the EOS parameters. The classical mixing rules of van der Waals (vdW) have already been tested for polymer solvent mixtures, however, it has been observed that, in order to fit the experimental data, some unrealistic values are necessary for the binary interaction

The use of equations of state in phase equilibrium modeling instead of activity coefficient models is mainly a result of the recent development of a class of mixing rules that enable the use of liquid activity coefficient models in the EOS formalism. The implication of this change is far-reaching as an EOS offers a unified approach in thermodynamic property modeling. With this approach, the applicability of simple cubic EOS has been extended to complex systems, such as polymeric systems, if coupled with the appropriate activity coefficient model. Therefore, there is much interest in mixture EOS models capable of describing higher degrees of nonideality than that possible with the van der Waals one-fluid

Future development of EOS for polymer mixtures is unclear and some contradictory statements can be found in the literature. Some authors indicate that cubic equations can be extended to correlate and predict VLE in polymer mixtures accurately. On the other hand, others state that, considering the complexity of this type of mixture, simplicity is not a necessary requirement for an EOS, as the calculation of parameters for the mixture components is more important. There is agreement, however, on the fact that future The most apparent progress toward EOS with the ability to describe phase behavior with polymers has been made by applying statistical mechanics. Some early models derived from statistical thermodynamics assumed molecules to be arranged in a lattice, whereas many of the more recent theories picture molecules to be moving freely in continuous space. In lattice models, the molecules are assumed to have one or more segments, and the partition function of the system can be obtained by counting the possible configuration when these segments are arranged in hypothetical cells which are like the lattices in solid materials. Then the thermodynamic quantities can be calculated from the partition function on the basis of statistical mechanics.

A huge amount of work has been done on the understanding of phase behavior in polymeric mixtures, either from an experimental or theoretical point of view. As well as supplying important data, experiments enable the evaluation of EOS models for the correlation and/or prediction of phase behavior. A model, on the other hand, takes much less experimental effort and can guide the researcher/analyst in the right direction.

A detailed review of the different lines of developing equations of state for the calculation of fluid phase equilibria is given by [1]. Recently [2] presented and discussed in depth both classical and novel thermodynamic models, which have been developed and can potentially be used for industrial applications. A review of the use of some equations of state (EOS) for LDPE process simulation can be found in Orbey *et al.* [3] and Valderrama [4].

Although there have been some analyses on equations of state that can describe the phase equilibria involving polymers, additional assessments are necessary. In general, the available works concern a specific approach, not taking into account others. In addition, these reviews and surveys focus on detailed model theory or theoretical possibilities of model variations, with a few quotes from practical applications. This chapter therefore presents an overview of the progress on EOS models for polymer systems considering the following approaches:

In EOS applications only works dealing with phase equilibrium are discussed, other types of applications, such as solvent absorption and/or polymer swelling, are not addressed. The timeline diagram in Figure 1 shows some of the key developments and outstanding papers related to the development of equations of state for polymer systems, which are discussed in this chapter. The following notation is used:

**Figure 1.** Timeline of some Key Model Developments Addressed in this Chapter from van der Waals (vdW) Equation in 1873.

## **2. Cubic equations of state and mixing rules**

The first group of models to describe the phase behavior (by calculating the equilibrium constant) corresponds to the van der Waals equations of state, known as cubic equations, in either the original version or variants thereof. They are extremely simple and efficient for experimental data correlation. In this group, modifications of the Redlich-Kwong equation stand out, especially the Soave-Redlich-Kwong (SRK) [31] and the Peng-Robinson (PR) [32], which can calculate, often successfully, the vapor-liquid equilibrium for normal fluid and mixtures. However, application of the cubic equations of state for polymeric blends is not immediately obvious as this application does not follow standard procedures. The conventional method for calculating the pure parameters in cubic equations of state requires components' critical properties and vapor pressure, which do not exist for polymers. Therefore, two basic issues should be addressed when extending cubic equations of state for polymers and their mixtures. The first, presented in the following paragraphs is the description of the parameters of pure components, and the second is the choice of the mixing rule, which will be discussed later in sections 2.1-2.3.

There are four conditions to be satisfied when selecting the pure component parameters of a cubic equation of state. First, a polymer is non-volatile and therefore should not exhibit any vapor pressure. If there are oligomers in the mixture though, low vapor pressures might be considered. Therefore, critical properties may be assigned for the oligomers, treating them as conventional components. The second condition is that the equation of state should predict densities of molten polymers. The third condition requires that the parameters reflect the polymers' basic characteristics such as the degree of polymerization. This is important because experimental data demonstrate that these polymer characteristics directly affect the vapor-liquid equilibrium in polymer-solvent mixtures. The fourth point, somewhat connected to the third, requires easily accessible and physical meaning characteristics as input parameters for calculating the parameters of the equation of state. As stated before, the SRK equation of state is expressed by:

$$P = \frac{R \cdot T}{V - b} - \frac{a \{ T \}}{V \cdot \left( V + b \right)} \tag{1}$$

and the PR equation is given by:

360 Polymerization

 Cubic Equations of State [Huron and Vidal (HV) Mixing Rule (MR); Sako-Wu-Prausnitz (SWP) Equation; Wong and Sandler (WS) MR; vdW Applied to Polymer Solutions (vdW-P) by Kontogeorgis, Harismiadis, Fredeslund and Tassios; Linear Combination of Vidal and Michelsen (LCVM) Mixing Rules by Boukouvalas, Spiliots, Coutsikos,

 Lattice Models [Flory-Huggins (FH); Sanchez and Lacombe (SL) equation; key Modifications and Applications (MA) of SL by KIeintjens and KoningmeId, Panayiotou and Vera, Kiran and Xiong and Zhuang, Koak and Heidemann, Gauter and

 Perturbation theory [Wertheim Thermodynamic Perturbation Theory (TPT); Statistical Associating Fluid Theory (SAFT) by Chapman, Gubbins, Jackson, Radosz; Chen and Kreglewski [5] SAFT (CK-SAFT) by Huang and Radosz; Perturbed-Chain SAFT (PC-SAFT) by Gross and Sadowski; Simplified PC-SAFT (sPC-SAFT) by von Solms,

**Figure 1.** Timeline of some Key Model Developments Addressed in this Chapter from van der Waals

The first group of models to describe the phase behavior (by calculating the equilibrium constant) corresponds to the van der Waals equations of state, known as cubic equations, in either the original version or variants thereof. They are extremely simple and efficient for experimental data correlation. In this group, modifications of the Redlich-Kwong equation stand out, especially the Soave-Redlich-Kwong (SRK) [31] and the Peng-Robinson (PR) [32], which can calculate, often successfully, the vapor-liquid equilibrium for normal fluid and mixtures. However, application of the cubic equations of state for polymeric blends is not immediately obvious as this application does not follow standard procedures. The conventional method for calculating the pure parameters in

Tzouvaras and Tassios; Zhong and Masuoka (ZM) MR]

Heidemann, Krenz and Heidemann and de Loos]

**2. Cubic equations of state and mixing rules**

Michelsen and Kontogeorgis]

(vdW) Equation in 1873.

$$P = \frac{R \cdot T}{V - b} - \frac{a \{ T \}}{V \cdot \{ V + b \} + b \cdot (V - b)} \tag{2}$$

where *T* is temperature, *V* is molar volume, *P* is pressure and *R* is the universal gas constant. The parameter *a* is a measure of the attractive forces between molecules, and the parameter *b* is the co-volume occupied by these molecules.

The first attempt to apply a cubic equation to polymers was made by Sako *et al*. [7] in order to calculate the high pressure vapor-liquid equilibrium for a polyethylene-ethylene system. To overcome the problem of calculating the pure parameters for the polymer, Sako *et al*. [7] calculated the attractive parameter in the SRK equation using the London dispersion formula, and extrapolated the co-volume (*b*) values from n-alkane data. To take into account external degrees of freedom, they also added a third parameter *c*, whose values were fitted from density data. The SWP (Sako-Wu-Prausnitz) equation is a member of the cubic family:

$$P = \frac{R \cdot T \cdot (V - b + b \cdot c)}{V \cdot (V - b)} - \frac{a \begin{pmatrix} T \\ \hline V \cdot \end{pmatrix}}{V \cdot \begin{pmatrix} V + b \end{pmatrix}} \tag{3}$$

The SWP equation was used with relative success by Tork *et al*. [33] when calculating the phase equilibria in binary and ternary systems of polyolefins. The calculations focused on the high pressure phase equilibrium for ethylene-polyethylene systems and for liquidliquid equilibrium in systems containing either high density polyethylene or polyethylene-polypropylene copolymer. The results for the copolymer-solvent system were compared with those provided by the SAFT (Statistical Associating Fluid Theory) equation. The two equations of state can describe the UCST (Upper Critical Solution Temperature) and LCST (Lower Critical Solution Temperature) behavior as well as U-LCST, with similar precision. When using the SAFT equation, the binary interaction parameter is maintained constant, while in the SWP equation this parameter is expressed as a function of temperature. In addition, Sako *et al*. [7] investigated the influence of an inert gas on the LCST for the polyethylene-hexane system. The polydispersity of different polyethylene resins is considered when computing the phase equilibrium using pseudocomponents, chosen using the moments of experimental molecular weight distributions.

Kontogeorgis *et al*. [9] used the van der Waals equation of state to correlate vapor-liquid equilibrium data of polymer solutions. They proposed a method to calculate the interaction parameter *a* and co-volume *b* in the equation of state for polymers from two volumetric datasets at low pressure. Both parameters *a* and *b* (assumed to be independent of temperature) can be analytically expressed from two experimental molar volumes, each one at a different temperature. The pressure in the van der Waals equation is then considered equal to zero. The parameters *a* and *b* are linear functions of molecular weight. When dealing with polymer solutions, these parameters are obtained from van der Waals mixing rules as well as from the classic combination rules. Fitting only one binary parameter, the van der Waals equation of state is able to correlate the equilibrium pressure for various solutions of polyethylene and polyisobutylene accurately. However, large negative values for the binary interaction parameters, very different from typical values, are frequently required, indicating that this procedure, although empirically successful, does not have a significant physical basis. For almost athermic solutions, the Berthelot combining rule [34] is considered, and the binary interaction parameter is predicted by a simple function of the molecular weight of the solvent. Thus, satisfactory results are obtained.

The performance of cubic equations of state is directly related to the efficiency of mixing rules to represent the phase equilibria at high pressures. Basically, the mixing rules can be divided into two classes: van der Waals-type and those that incorporate excess Gibbs energy (*GE*).

#### **2.1. van der Waals mixing rules**

In order to extend the application of PR and SRK cubic equations of state for polymersolvent systems, the conventional mixing rules employed are those from van der Waals (vdW) [34], which are expressed as:

A Survey of Equations of State for Polymers 363

$$a = \sum\_{i} \sum\_{j} \boldsymbol{\chi}\_{i} \cdot \boldsymbol{\chi}\_{j} \cdot a\_{ij} \tag{4}$$

$$b = \sum\_{i} \sum\_{j} \boldsymbol{\chi}\_{i} \cdot \boldsymbol{\chi}\_{j} \cdot b\_{ij} \tag{5}$$

where *xi* and *xj* are the mol fractions, *aij* is the cross-energy parameter and *bij* is the cross covolume parameter.

It should be stressed that these rules are limited to non-polar fluids and therefore are unable to represent the highly non-ideal behavior of polar or associative fluids.

An empirical approach to overcome the shortcomings of the vdW mixing rule has been to simply add new parameters and composition dependence to the combination rule for parameter *a*, usually keeping the combination rule for parameter *b*. Some examples may be cited: Adachi and Sugie [35]; Panagiotopoulos and Reid [36] and Schwartzentruber *et al*. [37]. These modified rules usually aim to solve specific problems, and the number of binary interaction parameters is quite variable. There are many problems associated with these multiparametric combination rules which limit their use in process design for mixtures containing many components (such as mixtures of isomers). Among them, the dilution effect may be cited: as the number of components in a mixture increases, the molar fraction of any component becomes smaller. This leads to small contributions of the new added parameters and terms that are strongly composition dependent. Consequently, as the number of components increases, the mixing rule is effectively reduced to a quadratic dependence, as in the one-fluid van der Waals fluid theory [38].

#### **2.2. Mixing rules for excess free energy (GE) models**

362 Polymerization

distributions.

results are obtained.

**2.1. van der Waals mixing rules**

(vdW) [34], which are expressed as:

(*GE*).

The SWP equation was used with relative success by Tork *et al*. [33] when calculating the phase equilibria in binary and ternary systems of polyolefins. The calculations focused on the high pressure phase equilibrium for ethylene-polyethylene systems and for liquidliquid equilibrium in systems containing either high density polyethylene or polyethylene-polypropylene copolymer. The results for the copolymer-solvent system were compared with those provided by the SAFT (Statistical Associating Fluid Theory) equation. The two equations of state can describe the UCST (Upper Critical Solution Temperature) and LCST (Lower Critical Solution Temperature) behavior as well as U-LCST, with similar precision. When using the SAFT equation, the binary interaction parameter is maintained constant, while in the SWP equation this parameter is expressed as a function of temperature. In addition, Sako *et al*. [7] investigated the influence of an inert gas on the LCST for the polyethylene-hexane system. The polydispersity of different polyethylene resins is considered when computing the phase equilibrium using pseudocomponents, chosen using the moments of experimental molecular weight

Kontogeorgis *et al*. [9] used the van der Waals equation of state to correlate vapor-liquid equilibrium data of polymer solutions. They proposed a method to calculate the interaction parameter *a* and co-volume *b* in the equation of state for polymers from two volumetric datasets at low pressure. Both parameters *a* and *b* (assumed to be independent of temperature) can be analytically expressed from two experimental molar volumes, each one at a different temperature. The pressure in the van der Waals equation is then considered equal to zero. The parameters *a* and *b* are linear functions of molecular weight. When dealing with polymer solutions, these parameters are obtained from van der Waals mixing rules as well as from the classic combination rules. Fitting only one binary parameter, the van der Waals equation of state is able to correlate the equilibrium pressure for various solutions of polyethylene and polyisobutylene accurately. However, large negative values for the binary interaction parameters, very different from typical values, are frequently required, indicating that this procedure, although empirically successful, does not have a significant physical basis. For almost athermic solutions, the Berthelot combining rule [34] is considered, and the binary interaction parameter is predicted by a simple function of the molecular weight of the solvent. Thus, satisfactory

The performance of cubic equations of state is directly related to the efficiency of mixing rules to represent the phase equilibria at high pressures. Basically, the mixing rules can be divided into two classes: van der Waals-type and those that incorporate excess Gibbs energy

In order to extend the application of PR and SRK cubic equations of state for polymersolvent systems, the conventional mixing rules employed are those from van der Waals Like conventional phase equilibrium modeling, there are two basic modeling tools for dealing with polymer-solvent mixtures: excess Gibbs free energy (*GE*) models (or activity coefficient models) and equation of state models. There are plenty of models in each category and selecting the best model for a specific project can often be quite difficult. Furthermore, equation of state and activity coefficient models have varying abilities in extrapolating data beyond given ranges of temperature and pressure, which further hampers the choice of the best model. These models also behave differently when predicting vapor-liquid equilibrium from other measured properties, such as the infinite dilution activity coefficient for a polymer in solvent.

Over the last two decades several methods combining activity coefficient models with equations of state have emerged. These methods are useful for correlating/predicting the phase equilibria of conventional mixtures, and are promising for mixtures containing polymers. Moreover, they allow us to investigate an activity coefficient model in two ways: first as a conventional model (i.e. in the approach *γ*-*φ*); secondly as part of an equation of state. In general, activity coefficient models are considered more flexible to accommodate a highly complex phase behavior. Equation of state models, on the other hand, may take into account the effects of compressibility in a thermodynamically more consistent way, and are most useful at higher pressures.

Huron and Vidal [6], pioneers in this field, incorporated an excess Gibbs free energy model into a mixing rule. Their method is based on three assumptions: (i) the excess Gibbs free energy, calculated from an equation of state at infinite pressure equals the excess Gibbs free energy calculated from an activity coefficient model for the liquid phase (ii) the co-volume parameter *b* is set to the volume at infinite pressure (iii) the excess volume equals zero. Thus the mixing rule is written as:

$$
\alpha\_V = \frac{1}{q\_2} \cdot \frac{G^E}{\mathbf{R} \cdot T} + \sum\_i \mathbf{x}\_i \cdot a\_i \tag{6}
$$

where ( ) *<sup>V</sup> a bRT* and <sup>2</sup> *q* ln 2 for SRK equation. Huron and Vidal [6] showed that their mixing rule gives good results for non-ideal mixtures. Soave [39] showed that the Huron and Vidal rule represents an improvement over the classical quadratic mixing rules and can accurately correlate the vapor-liquid equilibrium for highly nonideal systems. The Huron and Vidal mixing rule has also been applied to several polar and asymmetric systems [40]-[42].

The Huron and Vidal mixing rule has some undesirable characteristics, such as: it does not reproduce the quadratic dependence of the second virial coefficient (QDSVC) with the composition at low pressure; it has no predictive value because the parameters of the activity coefficient model, estimated at low pressure, have to be re-estimated at high pressure; furthermore, these parameters are temperature dependent. Various proposals [43]- [44] have tried to cope with these constraints, however, they fail to succeed as discussed below.

Mollerup [45] suggested an alternative method to that of Huron and Vidal, assuming that the excess volume is zero at low pressure and that the excess Gibbs free energies calculated from an equation of state and from an activity coefficient model can be matched in this condition. Therefore, the activity coefficient parameters do not need be re-estimated if pressure and temperature conditions correspond to those at which they are fitted. However, since this theory cannot be applied to supercritical fluids, as well as the difficulty of computing roots (of the equation of state) for the liquid phase at zero pressure, its application is restricted.

Heidemann and Kokal [46], in accordance with Mollerup [45], also take the reference state at null pressure, attempting, however, to overcome the problem of calculating the root for the liquid phase at zero pressure. The major contribution of this method is to propose an extrapolation procedure from the system pressure, enabling calculation at temperatures near and above the critical point. Important to mention is that this method requires the solution of a transcendental equation when calculating the mixing rule. Comparative studies demonstrate a better performance of the Heidemann and Kokal rule when compared to the Huron and Vidal rule [6].

A method very similar to Heidemann and Kokal's [46] was proposed by Michelsen [47]. The main difference between them lies in the extrapolation method used for supercritical components. The mixing rule in the Michelsen approach also requires the solution of a transcendental equation. The SRK-Wilson model (using SRK equation of state and Wilson activity coefficient model), obtained using this method, was tested to obtain the phase envelope, including the critical points, and to calculate the phase diagrams at high pressure, without re-estimating the parameters of the Wilson model. Good results were achieved.

Michelsen [48] modified his own method, considering an explicit mixing rule, i.e. avoiding the solution of the transcendental equation. The only drawback of this modification lies in the impossibility of ensuring the accurate reproduction of the *GE* model at low temperatures. Thus, imposing a linear mixing rule to the parameter *a*, an expression similar to the Huron and Vidal rule [6] was obtained, and therefore it is called the first-order Modified Huron and Vidal (MHV1) mixing rule.

Dahl and Michelsen [49] found out that replacing the linear approach (MHV1) by a quadratic approximation considerably improves the reproductibility of the *GE* model. The resulting mixture rule, a second-order modification of the Huron and Vidal rule, became known as the MHV2 (second-order Modified Huron and Vidal mixing rule), where the linear rule for the co-volume *b* was maintained. Like the Huron and Vidal mixing rule, the MHV2 does not satisfactorily describe the excess molar volume. Additionally, it is theoretically incorrect at the lower bound pressure (when the pressure goes to zero), where it does not show the QDSVC with the mole fraction.

Attempting to straighten out the theoretical inconsistency of the aforementioned mixing rules, Wong and Sandler [8] proposed a new method in which the rules fulfill the QDSVC with the composition at low pressure condition. The basic idea was to consider the excess Helmholtz free energy as much less dependent on pressure than the excess Gibbs free energy. In this way, the excess Helmholtz free energy at high pressure might be equal to the excess Gibbs free energy at low pressure. Therefore, this mixing rule is given by:

$$a = b \cdot \left(\sum\_{i} \frac{\boldsymbol{x}\_{i} \cdot \boldsymbol{a}\_{i}}{b\_{i}} + \frac{G^{E}}{c}\right) \tag{7}$$

where:

364 Polymerization

where

[40]-[42].

below.

application is restricted.

Huron and Vidal rule [6].

most useful at higher pressures.

the mixing rule is written as:

account the effects of compressibility in a thermodynamically more consistent way, and are

Huron and Vidal [6], pioneers in this field, incorporated an excess Gibbs free energy model into a mixing rule. Their method is based on three assumptions: (i) the excess Gibbs free energy, calculated from an equation of state at infinite pressure equals the excess Gibbs free energy calculated from an activity coefficient model for the liquid phase (ii) the co-volume parameter *b* is set to the volume at infinite pressure (iii) the excess volume equals zero. Thus

their mixing rule gives good results for non-ideal mixtures. Soave [39] showed that the Huron and Vidal rule represents an improvement over the classical quadratic mixing rules and can accurately correlate the vapor-liquid equilibrium for highly nonideal systems. The Huron and Vidal mixing rule has also been applied to several polar and asymmetric systems

The Huron and Vidal mixing rule has some undesirable characteristics, such as: it does not reproduce the quadratic dependence of the second virial coefficient (QDSVC) with the composition at low pressure; it has no predictive value because the parameters of the activity coefficient model, estimated at low pressure, have to be re-estimated at high pressure; furthermore, these parameters are temperature dependent. Various proposals [43]- [44] have tried to cope with these constraints, however, they fail to succeed as discussed

Mollerup [45] suggested an alternative method to that of Huron and Vidal, assuming that the excess volume is zero at low pressure and that the excess Gibbs free energies calculated from an equation of state and from an activity coefficient model can be matched in this condition. Therefore, the activity coefficient parameters do not need be re-estimated if pressure and temperature conditions correspond to those at which they are fitted. However, since this theory cannot be applied to supercritical fluids, as well as the difficulty of computing roots (of the equation of state) for the liquid phase at zero pressure, its

Heidemann and Kokal [46], in accordance with Mollerup [45], also take the reference state at null pressure, attempting, however, to overcome the problem of calculating the root for the liquid phase at zero pressure. The major contribution of this method is to propose an extrapolation procedure from the system pressure, enabling calculation at temperatures near and above the critical point. Important to mention is that this method requires the solution of a transcendental equation when calculating the mixing rule. Comparative studies demonstrate a better performance of the Heidemann and Kokal rule when compared to the

*G*

1 *<sup>E</sup> V i i i*

2

( ) *<sup>V</sup> a bRT* and <sup>2</sup> *q* ln 2 for SRK equation. Huron and Vidal [6] showed that

*x*

*q RT* (6)

$$b = \frac{\sum\_{i} \sum\_{j} \boldsymbol{\alpha}\_{i} \cdot \boldsymbol{\alpha}\_{j} \cdot \left(b - a/R \cdot T\right)\_{ij}}{1 - \sum\_{i} \boldsymbol{\alpha}\_{i} \cdot \left(a\_{i}/b\_{i} \cdot R \cdot T\right) - \left(G^{E} \not\subset R \cdot T\right)}\tag{8}$$

$$\frac{1}{2}\left(b - a\big/\mathbb{R}\cdot\boldsymbol{T}\right)\_{\boldsymbol{i}\boldsymbol{j}} = \frac{1}{2}\cdot\left[\left(b - a\big/\mathbb{R}\cdot\boldsymbol{T}\right)\_{\boldsymbol{i}} + \left(b - a\big/\mathbb{R}\cdot\boldsymbol{T}\right)\_{\boldsymbol{j}}\right]\cdot\left(1 - K\_{\boldsymbol{i}\boldsymbol{j}}\right) \tag{9}$$

and *c* is a constant, equals –ln(2) for the SRK equation. The mixing rule gives the correct low density limit (the mixture second virial coefficient has a quadratic dependence on mole fraction). The high density limit, in turn, is consistent with experimental data: the equation of state provides the same molar excess Helmholtz free energy at infinite pressure as a function of composition, as well as that obtained from the selected activity coefficient model. The Wong and Sandler [8] mixing rule is not density dependent. It should be highlighted that, unlike the methods proposed above, the Wong-Sandler mixing rule introduced an additional binary interaction parameter besides those predicted by the activity coefficient model, which is the second virial coefficient binary interaction parameter itself. According to the authors, this parameter as well as those from the activity coefficient model, estimated at low pressure, may be interchangeably used (i.e. without restriction) at high pressure.

Boukouvalas *et al*. [10] proposed a new mixing rule for the parameter *a* in the attractive term of cubic equations of state. The idea was to make a linear combination between the Huron and Vidal and the MHV1 mixing rules, producing the name LCVM (Linear Combination of Vidal and Michelsen mixing rules), which may be expressed as:

$$
\alpha = \mathbb{X} \cdot \alpha\_{\vee} + \left(1 - \mathbb{X}\right) \cdot \alpha\_{M} \tag{10}
$$

where *<sup>V</sup>* and *<sup>M</sup>* are given by Vidal and Michelsen rules, respectively. The contributions related to *α* are weighted by a factor *λ*, which is proposed by Boulouvalas *et al*. [10] to be 0.36. For the parameter *b*, a classical linear rule was considered. The performance of this model was compared to MHV2 and MHV1 models, using the Soave equation and the UNIFAC [UNIQUAC (UNIversal QUAsiChemical) Functional-group Activity Coefficient] for nonpolar and polar systems, symmetrical and asymmetrical, low and high pressures. The results indicate an equivalent performance of LCVM compared to the other two models when investigating systems containing molecules of similar sizes. In mixtures composed of molecules with quite different sizes, in particular gas systems with alkanes, the LCVM rule showed superior results.

Zhong and Masuoka [50], based on experimental data, evaluated the MHV1 mixing rule with SRK equation of state and the original UNIFAC model for *GE*. They found out that: 1) SRK equation with MHV1 can not reproduce the *GE* of the *GE* model used in the mixing rule for asymmetric systems, even at low pressure; 2) the original UNIFAC is not accurate for asymmetric systems with large alkanes. The first point reflects the deficiency of MHV1 for asymmetric systems, while the second is caused by the low predictive ability of UNIFAC for systems containing large alkanes. Moreover, it is evident that, although MHV1 is able to reproduce the *GE* model exactly, it may not be as accurate for gas-large alkane systems as UNIFAC can not describe these systems properly. As a result, it is pointless to pursue exact reproduction of the *GE* model when attempting to improve the predictive capability for these systems. However, it is interesting to observe that the SRK equation with MHV1 can satisfactorily reproduce *GE* experimental data if a correction factor is added to *GE* in the original UNIFAC model used in MHV1. With this observation, a new mixing rule was proposed: the MR1 (mixing rule 1), obtained by Zhong and Masuoka [50]. The MR1 rule is very accurate for systems composed og gas and large alkanes, when the correction parameter is obtained as a simple correlation function of the carbon number for a given gas.

366 Polymerization

where *<sup>V</sup>* and

showed superior results.

and *c* is a constant, equals –ln(2) for the SRK equation. The mixing rule gives the correct low density limit (the mixture second virial coefficient has a quadratic dependence on mole fraction). The high density limit, in turn, is consistent with experimental data: the equation of state provides the same molar excess Helmholtz free energy at infinite pressure as a function of composition, as well as that obtained from the selected activity coefficient model. The Wong and Sandler [8] mixing rule is not density dependent. It should be highlighted that, unlike the methods proposed above, the Wong-Sandler mixing rule introduced an additional binary interaction parameter besides those predicted by the activity coefficient model, which is the second virial coefficient binary interaction parameter itself. According to the authors, this parameter as well as those from the activity coefficient model, estimated at low pressure, may be interchangeably used (i.e. without restriction) at high pressure.

Boukouvalas *et al*. [10] proposed a new mixing rule for the parameter *a* in the attractive term of cubic equations of state. The idea was to make a linear combination between the Huron and Vidal and the MHV1 mixing rules, producing the name LCVM (Linear Combination of

>

related to *α* are weighted by a factor *λ*, which is proposed by Boulouvalas *et al*. [10] to be 0.36. For the parameter *b*, a classical linear rule was considered. The performance of this model was compared to MHV2 and MHV1 models, using the Soave equation and the UNIFAC [UNIQUAC (UNIversal QUAsiChemical) Functional-group Activity Coefficient] for nonpolar and polar systems, symmetrical and asymmetrical, low and high pressures. The results indicate an equivalent performance of LCVM compared to the other two models when investigating systems containing molecules of similar sizes. In mixtures composed of molecules with quite different sizes, in particular gas systems with alkanes, the LCVM rule

Zhong and Masuoka [50], based on experimental data, evaluated the MHV1 mixing rule with SRK equation of state and the original UNIFAC model for *GE*. They found out that: 1) SRK equation with MHV1 can not reproduce the *GE* of the *GE* model used in the mixing rule for asymmetric systems, even at low pressure; 2) the original UNIFAC is not accurate for asymmetric systems with large alkanes. The first point reflects the deficiency of MHV1 for asymmetric systems, while the second is caused by the low predictive ability of UNIFAC for systems containing large alkanes. Moreover, it is evident that, although MHV1 is able to reproduce the *GE* model exactly, it may not be as accurate for gas-large alkane systems as UNIFAC can not describe these systems properly. As a result, it is pointless to pursue exact reproduction of the *GE* model when attempting to improve the predictive capability for these systems. However, it is interesting to observe that the SRK equation with MHV1 can satisfactorily reproduce *GE* experimental data if a correction factor is added to *GE* in the original UNIFAC model used in MHV1. With this observation, a new mixing rule was proposed: the MR1 (mixing rule 1), obtained by Zhong and

*<sup>M</sup>* are given by Vidal and Michelsen rules, respectively. The contributions

*V M* 1 (10)

Vidal and Michelsen mixing rules), which may be expressed as:

A new mixing rule for cubic equations of state, particularly suitable for highly symmetric systems, was proposed by Zhong and Masuoka [51]. It was validated by two cubic equations of state: a modification of Peng-Robinson equation proposed by Stryjek and Vera (PRSV) [52] and the SRK equation. As there is no critical point for polymers, parameters *a* and *b* in the equation of state cannot be calculated from critical properties and because polymers are almost non-volatile, their vapor pressures are very low. Therefore, it is possible to use zero-pressure experimental densities to determine them. Alternatively, the approach adopted by Orbey and Sandler [38] may be employed, i.e. to use densities at low pressure with a hypothetical and very low vapor pressure, for example 10-7 MPa. Zhong and Masuoka [51], in the proposal of this new mixing rule, considered a null Helmholtz free energy in the limit when pressure tends to infinity. This is the only difference between this mixing rule and the one proposed by Wong and Sandler [8]. As a result, only one parameter is necessary in the new mixing rule, which is much simpler than the method of Wong and Sandler [8], and is as simple as Kontogeorgis *et al*. method [9]. When using the new mixing rule for ten polymer solutions in a wide temperature range, the results show that it allows cubic equations of state to correlate the vapor-liquid equilibrium of polymer solutions precisely even if just one temperature independent parameter is used. These results verify that the assumption, namely the excess Helmholtz free energy is null at infinite pressure, is feasible, or at least acceptable, for polymer solutions. The authors demonstrate that accurate correlations for polymer solutions are insensitive to parameters *a* and *b* in the equation of state.

In recent years, many studies have focused on improving hybrid models, i.e. equations of state which embody *GE* models into mixing rules. They attempt to expand their applicability to more complex systems, such as those containing highly polar components or molecules with significantly different sizes (e.g., polymer-solvent), without loosing versatility and simplicity. Recently, Ahlers and Gmehling [53] proposed the VTPR model (Volume Translated Peng-Robinson) which brings together the UNIFAC and Peng-Robinson equation with translated volume. In the VTPR model, the two Flory-Huggins (FH) type combinatorial terms [12], which come from the equation of state and from the UNIFAC model, as well as the Staverman-Guggenheim [54] contribution of the UNIFAC combinatorial term, were eliminated. Moreover, an empirical approach in VTPR incorporates different exponents in the combination rule for the crossed co-volume of the equation of state, depending on the system studied: for those without polymer, it was set to 0.75, whereas for solvent-polymer systems, it was set to 0.5. This empirical approach, however, introduces some uncertainties. For example, it is not possible to set a single parameter for mixtures containing two solvents and a polymer. Furthermore, it is difficulty to choose two exponents for some systems containing a molecule similar to a polymer, for example, propane/hexacontane.