Mach‐Zehnder interferometry (MZI) was invented by Ludwig Zehnder in 1891  and was subsequently refined by Ludwig Mach in the year after . The techniques were invented taking advantages of the interference phenomena of light to measure the phase difference between the two light beams in which one is varied by the presence of a sample. As illustrated in Figure 1, the basic structure of the interferometry is composed of one beam splitter (half mirror) and two reflecting mirrors. The coherent light beam from a laser after collimation was divided into two beams: the reference beam and the test beam on which the sample is interposed. These two beams serve as two arms, the reference and the test arms of the interferometry. The presence of an object on the test arm will result in the difference in optical path length, thereby changing the interference pattern of the laser at the half mirror [HM2]. The fringe patterns can be monitored and recorded either along the direction of the reference beam or the test beam. Compared to other interferometers like Michelson, the separation of the two arms of MZI can provide a wide application due to large and freely accessible working space though the optical alignment is relatively difficult. Taking advantage of this spacious working place, MZI has been utilized for various experiments: electron interferometer functioning in high magnetic field , flow visualization and flow measurements , for sensing applications . Furthermore, optofluidic Mach‐Zehnder interferometer for sensitive, label‐free measurements of refractive index of fluids was also developed . The unique structure of Mach‐Zehnder has also been utilized for optical communication as a modulator . On the other hand, a lot of efforts have been made to fabricate microscale optical systems including Mach‐Zehnder interferometer modulators using polymeric materials [8, 9]. In this chapter, we focus on studies on the local deformation in polymeric systems undergoing photocuring by ultraviolet (UV) light. Since the polymer mixture undergoes transition from liquid to solid by the reaction and at the same time phase separation takes place, the deformation (shrinkage and/or swelling) would affect the phase separation process and the resulting morphology. Mach‐Zehnder interferometry would be useful to monitor the extent of deformation in the nanometer scales during the reaction.
2. Instrumentation and data analysis
The block diagram of the MZI used in our experiments is illustrated in Figure 2.
The details are described in a previous publication . Briefly, a coherent 632.8 nm light beam from a He‐Ne laser is passed through a Glan‐Thompson prism and is converted into vertically polarized light. The polarized light beam was collimated by using collimating lens before entering the basic unit of the Mach‐Zehnder interferometer. After passing the half mirror [HM1], incident light was divided into two beams, forming the reference arm and the test arm of the MZI. A sample was interposed on the test arm and was half hidden by a movable mask to produce the reference part on the sample. The deformation of the sample under curing is obtained from the variation of the part irradiated with UV light compared to the part hidden by a mask on the same sample. In order to observe the deformation caused by the curing reaction, the sample was submitted to the sequence of masking‐irradiating‐mask removing‐recording. Movement of this mask is controlled with the precision of a micrometer. Details of the operation are provided in . These two laser beams encounter with each other at the half mirror [HM2] to produce interference patterns which were recorded on a charge‐coupled device (CCD) camera. The data were subsequently analyzed using a computer.
2.1. Basic theory of light interference phenomena
Two monochromatic planar waves E1 and E2 traveling along the two arms of the Mach‐Zehnder interferometer can be expressed by the following wave equation in complex form:
Here E and ϕ represent respectively the amplitudes and the phases of the two waves and i is the imaginary number.
The phase of these two waves can be written as
where k and n0 are respectively the wavenumber of the incident light and the refractive index of the sample before curing. L1 and L2 are respectively the path length of light traveling along the reference and the test arms. The phase difference between the two beams can be expressed by
where n0, the refractive index of air was set equal unity. If the length of two arms is set equal , . There is no interference for this particular case. In the presence of a sample with refractive index n and the thickness d interposed on the test arm, the phase of the wave traveling along this direction becomes
The phase difference in the presence of the sample becomes:
For the case = , the phase difference becomes
Therefore, the optical path length (OPL) of the sample before irradiation is
In general, both the refractive index and the thickness of the sample are varied by the reaction:
Since both and are small, can be neglected, leading to the final result
For the case, the change in refractive index is negligible, , the optical path length can be approximately expressed as
If the initial thickness (before curing) of the sample is , from definition, the deformation (either shrinkage or swelling) is given by the below equation:
The OPLD on the left‐hand side can be obtained from MZI experiments. Therefore, if the change in refractive index before and after the curing reaction can be directly measured using some instrument like prism coupler , the change in the sample thickness can be obtained.
2.2. Data analysis
The interference patterns obtained for a polymer film under in situ photocuring on the test arm of the MZI are recorded by using a CCD camera. Though the laser beam was passed through the spatial filter to select the best part of the beam and was subsequently collimated before entering the MZI unit, the interference patterns are slightly affected by the spatial distribution of the laser intensity. This effect can be removed by performing some correction assuming that the shape of the laser beam is Gaussian .
The interferograms obtained before and after this correction for the intensity distribution of a He–Ne laser (NEC, 1 mW) in the case a polystyrene/poly(vinyl methyl ether) PS/PVME (30/70) blend was used as sample are, respectively, illustrated in Figures 3 and 4. To reduce noise, the 2D data (480 pixel × 640 pixel) were divided into 48 horizontal strips with the dimension (10 pixel × 640 pixel) for each strip. Data along the y‐axis for each strip were then averaged to provide 1D data as shown in Figure 3(a).
In general, the real part of the intensity of an interferogram is a periodic function of distance and can be expressed in 1D as follows:
where and are the amplitude and phase of the signal, respectively.
Finally, the OPLD can be obtained for the left‐hand side of Eq. (12).
Polymers used in this study are the derivatives of poly(ethyl acrylate) (PEA), polystyrene (PS) and poly(vinyl methyl ether) (PVME). The mixture of PS and PVME exhibits miscibility at room temperature, providing a system for studying the effects of shrinkage on phase separation of polymer blends. All polymers used here have molecular weight larger than 100.000 and the molecular weight distribution around 2.0. The details of chemical synthesis and sample characterization are described in previous publications [13, 14].
Samples for MZI studies were obtained by solvent casting method and were dried under vacuum at least one night. All the samples PS/PVME mixtures with the dimension (20 mm × 20 mm × 10 μm) were annealed under vacuum over 2 h at temperature above the glass transition temperature (Tg) of the blend to erase the thermal history of the preparation process.
3.1. Photodimerization of anthracene as a photocuring reaction
To photo‐cross‐link a polymer with UV irradiation, photosensitive anthracene was chemically labeled on a given polymer by copolymerizing its monomer with a photoreactive monomer by copolymerization. By doing so, photoreactive anthracene moieties were introduced into the polymer component under examination. The labeling content of anthracene can be adjusted by varying the ingredients of the coupling reactions. Upon irradiation with 365 nm UV light, anthracene undergoes photodimerization as illustrated in Figure 5 for the case of PEA chains.
3.2. In situ observation of the deformation kinetics in homopolymers undergoing photocuring and relation to physical aging of the photocured polymer
Poly(ethyl acrylate) (PEA, Mw = 1.6 × 105, Mw/Mn = 2.2) was prepared by conventional free radical polymerization. To be able to cure PEA with UV irradiation, the PEA was chemically labeled with anthracene which served as a cross‐linker of the PEA chains as illustrated in Figure 5. Upon irradiation with 365 nm UV light, the anthracene moieties labeled on PEA undergoing photodimerization, generating PEA networks in the sample. As a consequence, the sample gradually approaches the glassy state and exhibits shrinkage due to the liquid→solid transition. However, this shrinkage in this particular case is fairly small and could not be observed via monitoring the change in the sample thickness by laser‐scanning confocal microscopy as in the case of photopolymerization . As the cross‐link density in the sample reaches a critical value, PEA enters the glassy state. Depending on the rate at which PEA enters the glassy state, physical aging  could occur. This feature can be observed via the irradiation intensity dependence of the shrinkage associated with irradiation time as shown in Figure 6. Here, the deformation which was calculated from the OPLD data given in Eq. (13) for the case of negligible change in refractive index is plotted versus irradiation time and elapse time. It is worth noting that the physical aging phenomena are evidenced by the continuation of shrinkage after stopping irradiation. These results suggest that the sample with the cross‐link density γ ~ 2 already enters the glassy state during irradiation, exhibiting the physical aging phenomena. Compared to the result obtained at low light intensity, it was found that the physical aging of photo‐cross‐linked PEA sample emerges at the cross‐link density /chain. This aging process becomes stronger under irradiation with higher light intensity. From the plot of normalized shrinkage ( vs. non‐dimensionalized elapse time defined as () where is the elapse time and is the characteristic time of the aging process, it was found that all the aging data obtained with a constant irradiation intensity can be expressed by a master curve .
3.3. Local deformation of miscible polymer blends under photocuring and relation to physical aging
Mach‐Zehnder interferometry was also utilized to detect the local deformation in miscible polymer blends polystyrene derivative (PS) and poly(vinyl methyl ether) (PVME). The curing reaction was performed taking advantages of the photodimerization of anthracene chemically labeled on the PS chains. The curing reaction was followed by monitoring the change in the absorbance of the photo‐cross‐linker chemically labeled on PS. It was found that both the curing kinetics and the deformation induced by the curing reaction can be described by the Kohlrausch‐Williams‐Watts (KWW) kinetics for kinetically inhomogeneous systems :
where A and B are constant, and are, respectively, rate constant of the reaction and shrinkage. The KWW exponent and are less than 1 and in between 0.7 and 0.8.
From the kinetics data expressed by Eqs. (17) and (18), c can be obtained from MZI data and can be deduced from the cross‐linking data. It was found that there exists a strong correlation between the curing and the shrinkage processes as illustrated in Figure 7 .
However, the correlation between the cross‐link process expressed by the reduced cross‐link density and the deformation process indicated by the reduced strain cannot be well expressible by a master curve, particularly at high cross‐link density under high light intensity as shown in Figure 8. It is worth noting that and are the maximum value at which the cross‐link density and the deformation can be achieved at a given light intensity. The increase in the concentration fluctuations in the blends under curing, particularly under high light intensity, would be responsible for the deviation from the master curve.
From the data obtained by MZI, it was also found that the glass transition temperature also plays an important role in the deformation of the cured PS/PVME blends. From the in situ measurements of deformation by MZI under curing, it was found that the photocured sample undergoes shrinking during the irradiation process, but the sample also partially recovered by swelling back after stopping irradiation. This process is determined by the difference between the experimental temperature and the resulting glass transition temperature Tg of the cured sample at the time of stopping irradiation. This particular swelling behavior was observed upon raising the experimental temperature .
3.4. Local deformation observed by MZI in polymer blends undergoing phase separation in the bulk state
So far, Mach‐Zehnder interferometry has been utilized to monitor the local deformation in homopolymers and miscible polymer blends under photocuring. For cured polymer mixtures, phase separation took place as the reaction yield exceeds a critical value. The shrinkage induced by curing reaction not only affects the shape of the blend but also influences the phase separation process. For the polymer mixtures with a lower critical solution temperature (LCST) like PS/PVME, cross‐linking the PS component will enlarge the unstable region of the mixture and eventually lead to phase separation. The shrinkage of the mixture reveals some unexpected behavior shown in Figure 9, as an example, for a PS/PVME (20/80) blend photo‐cross‐linked by irradiation with 365 nm UV light. Here, the peak of the scattering intensity in situ monitored during the curing process appears and gradually moves toward the side of wide angle (large wavenumber q), suggesting that the length scale of the bi‐continuous structures resulting from the phase separation gradually decreases, instead of increase as in many cases observed for the conventional phase separation process . Taking into account that polymers often undergo shrinkage upon curing, the deformation of a PS/PVME (20/80) blend was in situ monitored by using a Mach‐Zehnder interferometry under irradiation with the same conditions. The period ξ of these spinodal structures induced by photocuring was calculated using the Bragg condition , where is the wavenumber corresponding to the scattering peak of the intensity distribution shown in Figure 9. The plot of the characteristic length scale of the morphology versus the net time of curing clearly shows that decreases with increasing curing (irradiation) time. This result is opposite to the conventional phase separation in the absence of shrinkage, that is, increases as phase separation proceeds. Obviously, as curing time increases, the characteristic length scale gradually decreases and finally reaches a stationary value, indicating the termination of the phase separation process. From the results obtained by Mach‐Zehnder interferometry, the elastic strain defined as () was calculated and its dependence on curing time was examined. It was found that the elastic strain () increases with increasing the irradiation intensity and significantly deviates from these decay curves when phase separation starts. The initial slope of the plot () vs. irradiation time, that is ) can be used as a measure of the characteristic time of shrinkage process induced by the curing reaction. The rate of shrinkage ki can be defined as . Figure 10 depicts the clear correlation between the shrinkage obtained from Mach‐Zehnder interferometry and the evolution of morphology detected by light scattering, demonstrating again the effectiveness of the Mach‐Zehnder interferometry.
A clear correlation was observed between the apparent rate of the phase separation obtained from light scattering experiment and the apparent rate of shrinkage obtained from MZI experiments . The results also reveal the significant roles of Mach‐Zehnder interferometry in the kinetic studies on polymer phase separation.
Besides the applications of MZI to the polymer researches described above, the effect of polymer molecular weight on the deformation of poly(ethyl acrylate) (PEA) was recently investigated during the photocuring process. The effects of the entanglement molecular weight of PEA on the shrinkage and swelling process were observed by MZI and the results are discussed in terms of polymer diffusion in entangled polymer networks .
4. Concluding remarks
For the applications in polymer research, Mach‐Zehnder interferometer (MZI) would be a simple instrument to in situ monitor the local deformation in the nanometer scales. The current MZI instrument can be improved in two aspects: accessibility to temperature dependence measurements and improvement of signal‐to‐noise ratio to increase the data precision. The former requires some careful temperature controls of the experimental environments around the sample and the MZI chamber. On the other hand, the later could be solved by introducing lock‐in detection of interference signals. These experiments are underway to prepare for the second generation of Mach‐Zehnder interferometry for research on photocuring of polymers.