Perspective pubs.acs.org/Macromolecules
Perspective of Positron Annihilation Spectroscopy in Polymers Y. C. Jean,*,† J. David Van Horn,† Wei-Song Hung,‡ and Kuier-Rarn Lee‡ †
Department of Chemistry, University of MissouriKansas City, 5110 Rockhill Road, Kansas City, Missouri 64110, United States Department of Chemical Engineering and R&D Center for Membrane Technology, Chung Yuan Christian University, Chung-Li, Taiwan 32023
‡
ABSTRACT: Positron annihilation spectroscopy (PAS) is a novel method that provides molecular-level information about complex macromolecular structure in a manner different from, but complementary to, conventional physical and chemical methodology. This paper presents a perspective of PAS in polymeric systems covering 12 aspects: historical, spacial, spherical quantum model, anisotropic structure, voids, positronium chemistry, time, positron annihilation lifetime spectroscopy and data analysis, variable monoenergetic slow positron beam techniques and depth profiling, elemental analysis, multidimensional instrumentation advances in PAS, and free volume and free-volume theories. related to the integrals of positron and electron wave functions.5,6 Uses and advancements of PAS in polymer science rely on a good understanding of the fundamental properties of Ps and of molecules and their interactions, the chemical and physical aspects of materials, and advances in instrumentation.
1. HISTORICAL ASPECTS The history of positron is still less than a century old. The positron is the antiparticle of the electron predicted by Dirac in 19301 and discovered by Anderson in 1933.2 The positronium (Ps), a bound state of the positron and an electron and the lightest atom, was observed in gases by Deutch in 1951.3 Soon after their discoveries, it was realized that the annihilation characteristics of positrons and Ps, such as lifetime and energy, contain information about the electronic properties within molecules and solids.4 During the past four decades, a novel method, positron annihilation spectroscopy (PAS), has been developed to determine atomic and molecular defects and interfacial properties of a wide variety of materials.5,6 PAS is scientifically very rich. It probes the most fundamental information on matter, i.e. wave functions, but it is still in the developing stage in terms of methodologies and technological applications. An ancient Chinese proverb says: Planting a good tree takes ten years, but building for the good of humanity takes a hundred years. An analogous example of a scientific discovery from the past century around the same time as the positron is nuclear magnetic resonance (NMR), discovered by Rabi in 1938.7 Today, NMR is one of the most important and powerful techniques used in science and medicine for human benefit. In this paper, we present a perspective of the use of PAS in the study of one form of complex matter, namely polymers, and hope this will motivate more researchers to apply PAS for humanity’s good for the future. In recent years, PAS has been successfully used to determine the free-volume, void, and layer properties in polymeric systems.8−46 The basic principle of using PAS in polymers is based on the fact that the positron and Ps are preferentially localized in pre-existing defects, including free volume (∼0.1−1 nm) and voids (>1 nm) in polymeric systems. The annihilation parameters (e.g., positron annihilation lifetime) are directly © 2013 American Chemical Society
2. SPACIAL ASPECT In the physical aspect, time and dimension are two of the most fundamental parameters to be considered in PAS. In terms of distance, positron annihilation takes place in a radius on the order of 10−13 m, according to the uncertainty principle for a photon’s momentum.6 In terms of molecular dimension, such as freevolume size, both positron and Ps localization have been experimentally observed47,8 and theoretically justified48,9 for hole sizes >0.1 nm. The trapping and tunneling of Ps in free volumes have been examined, and it is found that in common polymers Ps will spend its entire lifetime in a hole with radius >0.15 nm.48 Most polymers have sufficient free volume concentrations to trap Ps as the Ps diffusion length is ∼2 nm.48 This is due to the zero point energy resulting from Ps when it is localized in a finite dimension of an electronic environment. Localization is also due to the intrinsic small Ps atomic size, i.e., 0.159 nm.6 When Ps is localized in a hole, its effective size may be reduced due to the interacting potential of surrounding electrons to compress the Ps wave function inside the hole. A Ps size of 0.11 nm has been indicated from the molecular simulation of free volumes in polymers.49 Molecular and dynamic simulations provide the size of free volumes, thus providing an excellent avenue for understanding free-volume properties obtained by PAS and vice versa.22,49−52 Combining molecular and dynamic simuReceived: June 24, 2013 Revised: August 12, 2013 Published: August 19, 2013 7133
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where s is the eccentricity of ellipsoid [s = (a2 − b2)1/2/a], and the dimensions of semimajor and -minor axes (expressed in angstroms) are “a” and “b”, respectively. The modification for other structures of free volumes is an open area of research.57,58 A special PAS technique capable of determining the anisotropy of oriented polymers is angular correlation of annihilation radiation (ACAR) that measures coincident signals at small angular deviations from the expected 180° of 2γ radiation.5,6,8 This technique utilizes the measurement of PAS at the transverse direction of momentum density. On the basis of the same infinite potential spherical quantum model and the uncertainty principle, we have developed a relationship between the full width at halfmaximum of the ACAR spectra, θ1/2, and the free-volume radius, R, as8
lations with PAS experiments for the same polymer is an excellent approach to understand free-volume properties in polymers. The use of PAS to study the subnanometer regime is most promising since most conventional methods, including Xray techniques and electron microscopy, are more sensitive in the nano- to micrometer regions. It is noted that the localization of Ps free volumes is clearly known while the positron localization in free volumes of polymers deserves more investigations.
3. SPHERICAL QUANTUM MODEL For the theoretical understanding of PAS, a quantum mechanical model is suitable for PAS development because the positron and Ps are quantum particles and PAS probes the subnanometer regions and defects in polymers. Most existing models for PAS evaluation are based on quantum mechanics. A simple quantum mechanical model was proposed to derive a relationship between R (hole radius) and o-Ps lifetime by Tao in 1972.53 In this model, the o-Ps resides in a spherical well having an infinite potential barrier of radius R0 with a homogeneous electron layer ΔR in the region between the hole radius (R) and R0 (R0 = R + ΔR). This model has been popularly adopted and is generalized to both positron and o-Ps localization in a finite spherical hole. Such a model provides a simple relation between the positron or Ps lifetime, τ, and the mean free-volume radius (R). A semiempirical equation derived by fitting the measured oPs or positron lifetime (τ) in an infinite potential spherical model with known cavity sizes can be written in a general form as −1 ⎡ ⎛ 2πR ⎞⎤ R 1 ⎟⎥ + ⊗M sin⎜ τ = C ⎢1 − ⎝ R + ΔR ⎠⎦ ⎣ R + ΔR 2π
R = 16.60/θ1/2 − 1.656
where R and θ1/2 are expressed in Å and mrad, respectively. In polymeric materials, two-dimensional angular correlation of annihilation radiation spectroscopy (2D-ACAR)5 is particularly useful to determine the free-volume anisotropic structure. 2DACAR has been demonstrated to measure the two-dimensional shape of free volumes in oriented polymeric materials.59,60 2DACAR is a powerful method in determining the anisotropy of free volumes; however, the radiation damage to polymer samples under a strong positron source (>20 mCi or 7.4 × 10 8 Bq) is a concern since a good 2D-ACAR experiment takes several days. For crystalline or oriented polymeric systems, the 2D-ACAR method is still useful in determining free-volume and void structures. The challenge is the radiation damage of samples exposed to much higher doses of radiation. The use of slow positron beam techniques for 2D-ACAR may reduce this radiation damage problem.
(1)
In this model, the three parameters to be determined are C, ΔR, and ⊗M: the intrinsic positron/positronium lifetime, the electron layer thickness for the positron/positronium penetration to the wall region of R0, and an additional function generalized for the modification of cavity structure and the chemistry of polymers, respectively. For common polymers with an o-Ps component and in the absence of chemical interactions between molecules and positronium, ⊗M = 1, where τ = τ3 (the o-Ps pick-off lifetime) and C and ΔR are calibrated to be 0.5 ns and 1.656 Å, respectively.54,55 The parameter ⊗M, the modification function in eq 1, could be extended to at least three types of polymeric applications: free volumes with anisotropic structure, cavity size with R > 1 nm (voids), and molecules that strongly interact with o-Ps. Other theoretical models, such as finite potential well and classical models, could also be developed in the future along with this simple infinitive potential spherical quantum model.
5. VOIDS The same infinitive potential spherical model has been extended to include the 3γ process from o-Ps for hole sizes between 1 and 100 nm corresponding to o-Ps lifetimes longer than 10 ns.61,62 Equation 1 is replaced by expressing the Ps annihilation rate, λ (the reciprocal of lifetime, 1/τ), and expressed as61 ⎧ ⎛ ⎛ R − Ra ⎞b⎞ Ra ⎪ ⎜ ⎟ ⎟ + λ ⎜ 1 λ − ⎪ 2γ ⎜ 3γ (R ≥ Ra) ⎝ R + ΔR ⎠ ⎟⎠ λ=⎨ ⎝ ⎪ ⎪ λ 2Rγ + λ3γ (R < Ra) ⎩
(4)
where the two variables Ra and b were determined to be 0.8 nm and 0.55, respectively; λ2γ is the o-Ps pick-off annihilation rate (1/ τ3) of eq 1: ⊗M = 1, C = 0.5 ns, and ΔR = 1.656 Å Tao for radius R or Ra, respectively; λ3γ is the o-Ps 3γ annihilation rate.61 Equation 4 has been calibrated with o-Ps lifetimes observed in materials with known cavity sizes obtained by separate techniques. The above eq 4 is particularly useful in determining pore sizes in the nanometers range for industrial applications in semiconductors and specialty membrane evaluation.
4. ANISOTROPIC STRUCTURE The spherical quantum model can also be extended to nonspherical defect structures. For instance, one can consider ellipsoidal structures with semimajor and -minor axes, a, and b, respectively, with the same volume as a sphere in eq 1.56 This involves a numerical solution of Ps in an ellipsoid, and the results indicate that o-Ps lifetimes in ellipsoidal holes are shorter than that in a sphere having the same volume.56,57 The free-volume radius obtained by eq 1 from the measured o-Ps lifetime in an ellipsoidal shape is smaller than the real free volume in polymers based on spherical shape and can be modified by an equation M as ⊗M:56 M = (0.400s − 4.16s 2 + 2.76s 3)(l + 0.0018a)
(3)
6. POSITRONIUM CHEMISTRY Ps is the lightest atom that can chemically react with molecules and different functional groups by two processes: first, chemical quenching which decreases o-Ps lifetime (typically observed as τ3), and second, chemical inhibition which decreases the probability of o-Ps formation (typically observed as I3). Chemical quenching is caused by chemical reactions between Ps and
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Figure 1. Comparison of PAS with various techniques for the examination of defects and voids in materials (OM, optical microscopy; TEM, transmission electron microscopy; STM, scanning tunneling microscopy; AFM, atomic force microscopy; Mech, mechanical techniques).6
molecules, and chemical exhibition is related to the radiation chemistry of the positron in media. The chemistry of positron and Ps should be considered to enhance understanding of the data obtained from PAS. The chemical aspect of molecular interactions with the positron and Ps has been investigated by quantum and statistical mechanical calculations6,63,64 and by experimental measurements on positron annihilation rates.65,66 With recent advancements in computational and slow positron beam technology, a better understanding of PAS related to chemical information in polymers should be possible in the near future. Currently, theoretically calculated positron and Ps binding energies63,64 and annihilation rate65,66 information are compared with experimental PAS data mainly at the qualitative level, such as observing the general trends of Ps interacting with differing chemical entities. There is a lack of systematic investigation of PAS data with different functional groups in chemical structure. The modification of PAS parameters for chemical functional groups is one of the important areas for the positron and polymer scientists to pursue since the chemical information inside the surface of free volumes or voids is essentially unknown from existing methods and the positron and Ps are uniquely sensitive to inner surfaces where they are localized. Typical chemical function groups which exhibit strong quenching are nitroaromatics, quinones, maleic anhydride, and ions with redox potential 200 μm due to the energy of emitted positron (mean energy ∼200 keV). However, many important areas of polymeric applications require a variable monoenergetic slow positron beam (VMSPB), positrons with a well-controlled energy from a few eV to a few tens of keV, such that surface and multilayer structures can be analyzed. VMSPB was developed in the 1970s by the atomic and solid state physics community70,71 and was adapted by the polymeric community in the 1990s.77−92 By tuning the positron energy of VMSPB, one can define the mean implantation depth of interest in a polymeric sample based on an established equation.70,71 It is a powerful technique to determine the layer structure in a multilayer polymeric system by 7136
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developed to benefit more polymeric applications using PAS in the future. 9.1. Need of an Intense Positron Beam. An important development of VMSPB is the increase of positron beam’s brightness so that the microstructure of the surface and the interfacial problems of polymeric systems could be studied. This requires a high intensity slow positron beam. Currently, a beam diameter of
