PARAFAC Analysis for Temperature-Dependent NMR Spectra of Poly(Lactic Acid) Nanocomposite

This chapter provides a tutorial on the fundamental concept of Parallel factor (PARAFAC) analysis and a practical example of its application. PARAFAC, which attains clarity and simplicity in sorting out convoluted information of highly complex chemical systems, is a powerful and versatile tool for the detailed analysis of multi-way data, which is a dataset represented as a multidimensional array. Its intriguing idea to condense the essence of the information present in the multi-way data into a very compact matrix representation referred to as scores and loadings has gained considerable popularity among scientists in many different areas of research activities.


Introduction
This chapter provides a tutorial on the fundamental concept of Parallel factor (PARAFAC) analysis and a practical example of its application. PARAFAC, which attains clarity and simplicity in sorting out convoluted information of highly complex chemical systems, is a powerful and versatile tool for the detailed analysis of multi-way data, which is a dataset represented as a multidimensional array. Its intriguing idea to condense the essence of the information present in the multi-way data into a very compact matrix representation referred to as scores and loadings has gained considerable popularity among scientists in many different areas of research activities.
The basic idea of PARAFAC is so flexible and general that its application is not limited to a particular field of spectroscopy confined to a specific electromagnetic probe. Examples of the application include fluorescence (Christensen et al., 2005;Rinnan et al., 2005), IR (Wu et al., 2003), NMR (Bro et al., 2010), UV (Ebrahimi et al., 2008;Van Benthem et al., 2011) and mass spectroscopy (Amigo et al., 2008). The first part of this chapter covers the theoretical background of trilinear decomposition of three-way data by PARAFAC with comparison to bilinear decomposition of two-way data by Principal component analysis (PCA).
In the second part of this chapter, an illustrative example of PARAFAC analysis for threeway data obtained in an actual laboratory experiment is presented to show how PARAFAC trilinear model can be constructed and analyzed to derive in-depth understanding of the system from the data. Thermal deformation of several types of poly lactic acid (PLA) nanocomposites undergoing grass-to-rubber transition is probed by cross-polarization magic-angle (CP-MAS) NMR spectroscopy. Namely, sets of temperature-dependent NMR spectra are measured under varying clay content in the PLA nanocomposite samples. While temperature strongly affects molecular dynamics of PLA, the clay content in the samples also influences the molecular mobility. Thus, NMR spectra in this study become a three-way Now, let us consider another experiment with one more perturbation. As described above, stimulation of a single sample ends up with two-way data array. But what if we still have some more samples, whose properties (e.g. concentration) are different? We will repeat a similar experiment for every single sample. This generates multiple two-way data. Thus, the entire dataset eventually becomes a stuck of the multiple two-way data like a cube, which contains two dimensions concerning applied two perturbations. Such spectral dataset is PARAFAC Analysis for Temperature-Dependent NMR Spectra of Poly(Lactic Acid) Nanocomposite 291 described as a three-way array with the (i,j,k)th element denoting the spectral intensity value at the ith concentration, the jth temperature, and the kth wavenumber. For example, the samples will show the variation of their molecular structure depending on the temperature. This may be also influenced by the change in the concentration. Thus the spectral intensities of the samples are potentially influenced by the temperature as well as concentration.

PARAFAC model
It is possible to condense the essence of the information present in multi-way data into a very compact matrix representation referred to as scores and loadings. The basic hypothesis of factor analysis techniques is that the improved proxy of the original data matrix can be reconstructed from only a limited number of significant factors. Thus, while the score and loading matrices contain only a small number of factors, it effectively carries all the necessary information about spectral features and, eventually, it becomes possible to sorting out the convoluted information content of highly complex chemical systems. The detailed analysis of such matrices potentially brings useful insight into building a mechanistic model for understanding complex phenomena studied by spectroscopic method.
Principal component analysis (PCA) is mathematical decomposition of two-way data in terms of the orthogonal set of dominant factors, i.e., eigenvectors (Smilde et al., 2004;Shinzawa et al., 2010). Two-way data decomposition by PCA results in yielding two matrices called scores and loadings which complementarily represent the entire features broadly distributed in the two-way data as follows, where T and P are PCA score and loading matrices consisting of r vectors, respectively. The rank r corresponds to the number of principal components representing the significant portion of the information contained within the data matrix X. The selection of r is somewhat arbitrary. It is usually set to be a number, as small as possible but sufficiently large enough such that there are no obvious spectral features found in the residual matrix E PCA . The residual matrix E PCA is the portion of the original data, which is not accounted for by the first r principal components used for the data representation. The two matrices T and P complementally represent the entire features broadly distributed in X. Namely, T holds abstract information concerning the relationship among the samples and P contains summary of variable, e.g. wavenumber which provides chemically or physically meaningful interpretation to the pattern observed in T. For example, PCA of the two-way data based on temperature-dependent spectra provides T describing similar or dissimilar thermal behaviour of the sample during the perturbation period and corresponding P represent information on key molecular structure associated with such similar or dissimilar thermal behaviour of the sample.
For even more data, PARAFAC is used to decompose a multi-way data and Fig. 2 illustrates graphical representation of PARAFAC operation to decompose a three-way data into score and loading vectors. PARAFAC is utilized to decompose the multi-way data into a linear combination of score and loading matrices (Smilde et al., 2004;Bro, 2004). The information on behavior induced by the perturbations is effectively described by score vectors and corresponding loading vectors provide chemically or physically meaningful interpretation to the patterns observed in the scores of the PARAFAC trilinear model. Namely, by using PARAFAC operation, I×J×K array matrix X can be expressed in terms of a product of score and loading matrices, A, B, and C, and a residual matrix E as follows where (I × JK) refers to the way that the multi-way array is unfolded. The notation  means Khatri-Rao product which operate Kronecker product  on partitioned matrices defined as In PARAFAC analysis, the set of matrices A, B and C are usually obtained by iteratively solving alternating least-squares (ALS) problems

XA ( C B ) over A for fixed
B and C, as well as the minimization over B or C in the similar matrix operation manner under appropriate model constraints, such as the non-negativity of concentration and spectral intensity (Bro & de Jong, 1997;Bro & Sidiropoulos, 1998). General procedure of PARAFAC becomes as follows, Initialize B and C to obtain Z as A is given by www.intechopen.com PARAFAC Analysis for Temperature-Dependent NMR Spectra of Poly(Lactic Acid) Nanocomposite where the superscript + means the Moore-Penrose inverse. Then update Z as Update Z as C is given by If the residual between the original X and reconstructed X by Eq. 2 is greater than error criteria, one repeats Eqs. (4)-(9) until convergence.
The initial estimates for B and C is important to obtain sufficient minimization of the error criteria (Shinzawa et al., 2007(Shinzawa et al., , 2008a(Shinzawa et al., & 2008b. Although ALS algorithm usually offers an eventual convergence to the optimal solution with a sufficiently large number of iterations, it sometimes reaches the suboptimal local minimum (Jiang et al., 2003(Jiang et al., & 2004. Unfortunately, such local convergence does not usually offer a global minimum, but it may just be stuck in a local minimum, producing insufficient solution. The major cause of the suboptimal local convergence may be a poor initial estimation. One possible solution for this problem is to select proper initial estimate which is less sensitive to the presence of a local minimum, e.g. via signaler value decomposition (Bro & de Jong, 1997;Bro & Sidiropoulos, 1998;Wang et al., 2006;Awa et al., 2008).

PLA nanocomposite
A pertinent example for PARAFAC analysis based on NMR spectra of PLA nanocomposites is provided here to show how certain useful information can be effectively extracted from an actual laboratory experiment. Fig. 3 shows the molecular structure of PLA. PLA polymer is made up of many long chains consisting of the repeat unit shown in the figure. PLA is derived from renewable resources, such as corn starch via fermentation and it is biodegradable under the right conditions, such as the presence of oxygen (Tsuji et al., 2010). Thus, PLA is a possible candidate of a new class of renewable polymers as a substitute for the petrochemical polymers. However, the physical properties of PLA are inadequate for the replacement of conventional commodity plastics in many applications.
Nanocomposite is a technique to improve the physical strength, thermal resistance and gas barrier by the dispersion of nanoclay into the polymer (Katti et al., 2006). The improvement www.intechopen.com of such polymer properties by using nanocomposite is one of the primary areas of interest due to its potential applications. The polymer nanocomposites are generally formed by the addition of a small amount of nanoclay dispersion.  PLA nanocomposite samples used in this study were prepared with PLA (Teramac®, Unitika) and organically modified clay (S-BEN W®, Hojun). The samples were put into a Labo-plastomill consisting of a 30C150 kneader and an R100 mixer (Toyo Seiki Seisaku-sho, Ltd., Tokyo) to melt-blend at 190 °C and 50 rpm for about 10 minutes. Pellets thus obtained were pressed into 0.2 mm thick sheet sandwiched between two thick Teflon® films by a hot press at 190 °C. Fig. 5 represents the effect of nanocomposite on PLA probed by Thermomechanical analysis (TMA). TMA is a technique to monitor the physical deformation of object under a constant load, while varying the temperature. For example, in this case, the elongation of the PLA nanocomposite samples (clay content = 0, 5 and 15 wt%) were measured by imposing a 9.8 mN load, while varying the temperature from 35 to 140 °C at a rate of 10 °C per a minute. The elongation of the samples starts when the temperature reaches glass transition temperature (T g ) of PLA, i.e. approximately 60 °C (Zhang et al., 2010). Then it gradually www.intechopen.com PARAFAC Analysis for Temperature-Dependent NMR Spectra of Poly(Lactic Acid) Nanocomposite 295 increases with the increase of temperature and it finally reached constant levels at the close of the observation period, indicating that the observed plastic deformation is closely related to glass-to-rubber transition of the amorphous component of PLA. It is also noted that the samples results in the different levels of elongation depending on the clay content. For example, the neat PLA sample shows 14.4 % of elongation. In contrast, the PLAnanocomposite including 15 wt % of clay ends up with 9.1 % of elongation. The leveling off of the elongation indicates the formation of a network structure due to the presence of physical crosslinks created by the crystalline domain.
Although such observation effectively detects the macroscopic changes in the mechanical properties caused by the presence of clay particles, additional fundamental molecular level understanding of the reinforcement mechanism is also desired. Spectroscopic method should become an important tool to probe the phenomena at the molecular level.

PALAFAC analysis of NMR spectra of PLA nomocomposites
The temperature-dependent NMR spectra of the PLA samples collected under the varying temperature from 20 to 80 °C are shown in Fig. 6. Cross polarization-magic angle spinning (CP-MAS) NMR experiments were carried out on a Varian 400 NMR system spectrometer operated at 100.56 MHz for 13 C resonance with a cross polarization contact time of 2 ms (Fawcett, 1996). A zirconium oxide rotor of 4 mm diameter was used to acquire the NMR spectra at a spinning rate of 15 kHz. Each sample was packed into a 4 mm cylinder-type MAS rotor. A set of temperature-dependent NMR spectra were obtained under varying ambient temperature from 20 to 80 °C at every 20 °C step. The heating rate was approximately 10 °C per an hour.
Samples of semicrystalline polymers prepared from their melt possess complex supermolecular structure consisting of crystalline lamellae embedded in an amorphous www.intechopen.com Chemometrics in Practical Applications 296 matrix (Wunderlich, 1980). PLA essentially undergoes highly convoluted transition process, when temperature and its constitution are altered. These transitions include the melting of ordered molecular segments, as well as the grass-to-rubber transition and other relaxation of process of the amorphous component (Zhang et al., 2005;Meaurio et al., 2006).  The CP-MAS technique is ideal for the observation of 13 C spectra of solid samples. Since the local environment of a chemical group in solids are generally rigid, this leads to further considerations for crystallography or, more generally, molecular packing (Fawcett, 1996). The CPMAS NMR study of semicrystalline PLA samples is often complicated by the presence of overlapped contributions from coexisting crystalline and amorphous. For example, the unimodal peak observed around 69.5 ppm is assignable to CH structure which represents mobility of the main chain of the PLA (Tsuji et al., 2010;Kister et al., 1998). It is noted that the peak intensity gradually decreases with the increase of the temperature. This may be explained as the decrease in the cross polarization efficiency by the change in the molecular dynamics during the heating. Thus, the variation of the spectral intensity here reflects the structural alternation of PLA induced by the temperature.
More importantly, careful comparison of the samples reveals that the main feature of the NMR spectra of the three samples looks somewhat different. For example, the temperaturedependent NMR spectra of the PLA nanocomposite including 15 wt% clay provides specific three peaks at 70.5, 69.5 and 68.4, indicating the presence of the crystalline structure in the sample (Tsuji et al., 2010;Kister et al., 1998). When the sample has no clay in the system, PARAFAC Analysis for Temperature-Dependent NMR Spectra of Poly(Lactic Acid) Nanocomposite 297 these crystalline peaks are disappeared and compensated by the development of seemingly unimodal peak probably assigned to the amorphous of PLA (Tsuji et al., 2010;Kister et al., 1998). This indicates that the presence of the clay substantially influences supermolecular structure of the PLA. Consequently, it is very likely that the change in the spectral feature of the three-way data is closely related to temperature and clay content of the system. Thus, in turn, the fully detailed analysis of the data provides an interesting opportunity to probe the nature of the PLA nanocomposite by elucidating the variation of the NMR spectral intensity induced by the each perturbation with PARAFAC trilinear decomposition.

Score
Temperature, ºC Amorphous Crystalline Factor 2 Factor 1 Fig. 7. Score vectors in score matrix A representing thermal behaviours of amorphous and crystalline components in PLA samples. Fig. 7, 8 and 9 show results obtained from A, B and C matrices derived from PARAFAC analysis of the three-way NMR spectral data collected under varying temperature and clay content, respectively. Two major factors are indicated here, reflecting the fact that there are two species present in the system. One of the important benefits derived from PARAFAC decomposition of the multi-way data is the ability to rationally clarify the effect of the applied perturbations. For example, the matrix A represents abstract information on the temperature-induced behavior of the PLA under the influence of the clay content. In contrast, the matrix C holds essential information on the spectral intensity variation induced by the addition of the clay under the influence of the temperature. The matrix B contains loading vectors which provides chemical or physical interpretation to the patterns observed in the score matrices A and C.
It is noted that the loading vector of the first component of the matrix B (Fig. 8) resembles the spectral feature of the amorphous component of PLA. The loading vector of the second component of the matrix B shows characteristic three peaks assignable to crystalline component of the PLA. Thus it is most likely that the second factor represents thermal behaviours of the crystalline components in PLA samples.
Once the assignments for the loading vectors are established, it becomes possible to provide chemically meaningful interpretation to the score matrices A and C representing the dynamic behaviour of the components induced by the perturbations. For example, the score vector of the first factor in the matrix A represents the temperature-induced behaviour of the amorphous component of the PLA. On the other hand the score vector of the second factor means that of the crystalline component of the PLA. It is noted the score vector of the amorphous components exhibits obvious decrease with the temperature and such decrease becomes significant when the temperature exceeds its T g . In contrast, the change in the score value of the crystalline component is small, indicating no major variation during the heating process. The predominant variation of the amorphous component in the temperature region is explained as its glass-to-rubber transition. The change induced in the temperature region is associated with the Micro-Brownian motion of the PLA polymer segment. At a low temperature the amorphous regions of a polymer are in the glassy state. In this state the molecules are frozen on place. They may be able to vibrate slightly, but do not have any segmental motion. When the polymer is heated up to reach its T g , the molecules can start to wiggle around to become rubbery state. Such segmental motion predominantly occurs in amorphous region of PLA while such motion is strongly restricted in systematically folded crystalline lamellae structure. Thus, it is very likely the observed change of the amorphous is related to glass-to-rubber transition of the amorphous component.
Now it is important to point out again that the predominant elongation in the TMA occurred around T g . This elongation behaviour agrees well with the thermal behaviour of the amorphous component observed in the score matrix A. It thus suggests the physical elongation of the samples is essentially associated with the glass-to-rubber transition mainly occurred in the amorphous region.
It also becomes possible to provide the detailed interpretation to the pattern observed in the matrix C representing the clay-induced behaviours of amorphous and crystalline components in the PLA samples. The gradual decrease of the score of the first factor can be explained as the decrease of the amorphous component and the change in the sore of the first factor corresponds to the increase in the crystalline component by the addition of the This hypothesis is also clearly supported with corresponding transmission electron microscope (TEM) images and differential scanning calorimetry (DSC) results of the PLA nanocomposite sample. Fig. 10 represents the TEM images of the PLA nanocomposite sample including 15 wt% clay. For example, in Fig. 10(a), one can see that the clay is broadly distributed over the PLA matrix. On the other hand, Fig. 10(b) reveals that some parts of the interlayer gallery is obviously extended, suggesting the insertion of the PLA polymer into the clay layers, namely intercalation.
DSC curves of the PLA samples, represented in Fig. 11, clearly show the presence of glass transition temperature around 60 °C. It is important to point out that this glass-to-rubber transition of amorphous component agrees well with the change in the elongation observed in the TMA. More importantly, the samples also provide obvious crystallization peak around 110 °C. The crystallization peak shows gradual increase by the inclusion of the clay content, suggesting quantitative increase in the amount of the crystalline structure. Thus, it www.intechopen.com is very likely that the clay works as the nucleating agent to increase the frequency of the spontaneous nucleation of the PLA crystals.
All the results put together, it provides overall picture of the system. When the clay is dispersed in the PLA matrix, the PLA polymer located at interlayer or around surface layer of the clay develops crystalline structure more frequently. The generation of the crystalline structure of PLA is compensated by the decrease of the amorphous content. This should decrease the structural portion substantially undergoing glass-to-rubber transition above T g . Thus, in turn, it restricts the elongation of the samples during the heating process under a certain level of load. Consequently, it is demonstrated that PARAFAC analysis of the threeway data of the PLA nanocomposite samples effectively elucidates the mechanisms of the improvement of the mechanical property by the clay. By carrying out detailed band position shift analysis of the three way data of the temperature-and clay-dependent NMR spectra of the PLA samples, it becomes possible to extract chemically meaningful information concerning the variation of the crystalline structure closely associated with the nanocomposite system.  Fig. 11. DSC curves of neat PLA and PLA nanocomposite samples.

Conclusion
The basic background of PARAFAC and its practical example based on the temperaturedependent NMR spectra of the PLA nanocomposite samples are presented. The central concept of PARAFA decomposition of multi-way data lies in the fact that it can condense the essence of the information present in the multi-way data into a very compact matrix representation referred to as scores and loadings. Thus, while the score and loading matrices contain only a small number of factors, it effectively carries all the necessary information about spectral features and leads to sorting out the convoluted information content of highly complex chemical systems.
The effect of PLA nanocomposite is studied by the PARAFAC analysis of the temperaturedependent NMR spectra of several PLA nanocomposite samples including different clay contents. The PARAFAC analysis for the three-way data of the PLA nanocomposites revealed that the crystalline and amorphous structures of the PLA nanocomposites substantially undergo different transition under the heating. Namely, the change in the micro-Brownian motion of the polymer segments mainly occurs in the amorphous region when the PLA samples are heated up to their T g . It also revealed that clay potentially works as nucleating effect of the clay. Namely, it increases the frequency of the spontaneous nucleation of the PLA crystals. Thus, in turn, the change in the population of the rigid crystalline and rubbery amorphous provides the improvement of the physical property. Consequently, it is possible to derive in-depth understanding of the PLA nanocomposites.

Acknowledgment
A part of this work was financially supported by NEDO "Technological Development of Ultra-hybrid Materials" Project. www.intechopen.com