Polymer Physics (From Suspensions to Nanocomposites and Beyond) || Morphology of Free-Volume Holes in Amorphous Polymers by Means of Positron Annihilation Lifetime Spectroscopy

PART III

POSITION ANNIHILATION LIFETIMESPECTROSCOPY

391

10MORPHOLOGY OF FREE-VOLUMEHOLES IN AMORPHOUS POLYMERSBY MEANS OF POSITRONANNIHILATION LIFETIMESPECTROSCOPY

Giovanni Consolati and Fiorenza QuassoDipartimento di Fisica, Politecnico di Milano, Milano, Italy

10.1 Introduction

10.2 Experimental setup

10.3 Positronium in amorphous polymers

10.4 Estimation of hole sizes from o-Ps lifetime10.4.1 Tao–Eldrup equation10.4.2 o-Ps lifetime versus the cavity size in different geometries

10.5 Estimation of free-volume fraction

10.6 Discussion of some experimental results10.6.1 Polystyrene10.6.2 Oligomeric series

10.7 Conclusions

Polymer Physics: From Suspensions to Nanocomposites and Beyond, Edited by Leszek A. Utracki andAlexander M. JamiesonCopyright © 2010 John Wiley & Sons, Inc.

393

394 MORPHOLOGY OF FREE-VOLUME HOLES IN AMORPHOUS POLYMERS

10.1 INTRODUCTION

Various experimental techniques have been proposed for obtaining informationon the free volume in polymers (e.g., photoisomerization [Victor and Torkelson,1987], small-angle x-ray diffraction [Roe and Song, 1985], and photochromic labels[Lamarre and Sung, 1983]). Among the available techniques, positron annihilationlifetime spectroscopy (PALS) has become popular, since it is a relatively simple wayto probe the properties of free-volume holes in a nondestructive way [Mallon, 2003;Dlubek et al., 2004]. The technique is based on the fact that when positrons are injectedinto a material under investigation, some may form an unstable electron–positroncomplex, termed positronium (Ps), which becomes trapped in low-electron-densityregions of the host matrix (i.e., holes). In the ground state, Ps exists in two sublevels:ortho-Ps (o-Ps) and para-Ps (p-Ps), reflecting the different spin states of the particles(parallel and antiparallel, respectively). In a vacuum, o-Ps decays into three quantawith a lifetime of 142 ns. On the other hand, p-Ps decays into two quanta and hasa lifetime of 125 ps, that is, about three orders of magnitude smaller. In condensedmatter, o-Ps interacts with the electrons in the vicinity of the hole in which it istrapped, and its lifetime is determined primarily by the annihilations of the positronwith one of these external electrons in a relative singlet state [Jean, 1995] (a pick-off mechanism). This process greatly reduces the o-Ps lifetime, typically to valuesin the range 1 to 10 ns. Conversely, p-Ps is scarcely influenced by the presence ofmatter, due to its fast intrinsic annihilation rate. In the determination of free vol-ume by PALS, holes smaller than the Ps volume cannot be revealed; nevertheless,we note that such empty spaces are too small to be occupied by atoms, since thePs volume is equal to the volume of a hydrogen atom. Therefore, as a holes probe,Ps should not suffer from a significant low threshold in the sizes of holes detectedunless the number of holes undetectable by Ps is very large; in this case, these holesmay provide a contribution to the free volume [McCullagh et al., 1995; Olson et al.,2003].

Previous remarks suggest the existence of a correlation between o-Ps lifetimeand the sizes of the cavity, which can be cast in a quantitative form by suitablemodeling of the trapping site. The first model proposed was based on the simplestgeometry, the spherical hole [Tao, 1972; Eldrup et al., 1981]. This produces a mono-tonic relationship between o-Ps lifetime and hole radius and is used in most PALSinvestigations of polymers and porous materials (e.g., clays, cements). The sphericalmodel supplies results in qualitative agreement with expectations when the depen-dence of free volume on external stimuli is investigated (e.g., versus temperatureor mechanical stress). However, any model that assumes a particular hole shapegives a similar monotonic relationship [Jasinska et al., 1999]; therefore, it is notpossible to discriminate among different geometries of the cavity solely on the basisof variation of the void volume with the parameter investigated (e.g., temperature).

Here we show that it is possible to obtain information on hole morphology inpolymers, as well as changes in morphology due to external stimuli, by couplingPALS and dilatometry techniques and comparing the experimental results with thetheoretical evaluation of the free volume via the S-S equation [Simha and Somcynsky,

EXPERIMENTAL SETUP 395

1969]. First, to provide familiarity with the PALS technique, it is convenient to givea short description of the experimental setup and review the methodology by whichinformation on hole size is obtained.

10.2 EXPERIMENTAL SETUP

Positrons are generally obtained from 22Na, a radioisotope that emits a prompt: thatis, within 3 ps, a γ-ray with an energy of 1.274 MeV (Figure 10.1). This start signalmarks the birth of the positron. The stop signal is supplied by one of the annihila-tion photons (0.511 MeV); most annihilations ordinarily evolve into two γ-rays. Thesource strength is typically 0.04 to 0.8 MBq, and the source is prepared by depositinga droplet of an aqueous solution containing 22Na on a thin metallic foil or plasticsheet; after drying, the residue is covered by an identical support and sealed, to obtaina reusable source. A typical support is the polymide Kapton; its use is advantageoussince Ps does not form in it and subtraction of the positrons contribution to the totaltime annihilation spectrum is therefore simple [MacKenzie, 1983]. The source isinserted between two layers of the sample to be investigated (i.e., the sandwich con-figuration), whose thickness must be sufficient to stop 99.9% of the injected positrons(the range of positrons in matter from a 22Na source is about 170 mg/cm2 [Schraderand Jean, 1988]).

A schematic diagram of a PALS timing spectrometer is shown in Figure 10.2[Coleman, 2003]. It is formed by two detector channels, each consisting of a scin-tillator coupled to a photomultiplier tube (PMT). The scintillator converts the startor stop γ-ray into ultraviolet photons, which are absorbed by the window of thePMT, producing an emission of photoelectrons. Most commonly used scintillatorsare organic (plastic) or inorganic in nature (BaF2). The resulting photoelectrons aremultiplied by a series of intermediate dynodes to obtain a sufficient voltage pulse atthe output of the PMT anode; the resulting voltage is proportional to the energy of theγ-ray emitted by the scintillator. Discrimination between start and stop events occurson the basis of the different energies of the γ-ray detected: A constant fraction dis-criminator (CFD) on each channel of the spectrometer generates a fast timing signalwhenever a γ-ray with the correct energy is detected. A time-to-amplitude converter(TAC) enabled by the start signal from the CFD produces a voltage that increases

FIGURE 10.1 Decay scheme of 22Na.


TAC

Positronsource +Sample

ADC

PC

CFD CFD

PMTPMT

HV HV

DELAY

FIGURE 10.2 Positron annihilation lifetime spectrometer.

linearly with time, which stops at the arrival of the stop signal from the other CFD.The available signal at the output of the TAC is therefore proportional to the timeinterval elapsed between generation and annihilation of the positron. This signal isdigitized by a analog-to-digital converter (ADC) and transferred to the memory of apersonal computer (PC). Each memory channel corresponds to a voltage ∆v and con-tains the annihilation events occurring within a specified time interval ∆t. Linearityof the apparatus guarantees constancy of the ratio ∆t/∆v, which can be confirmedeasily through a calibration procedure.

Certain of the electronic units (CFD and TAC) have now been replaced by theirdigital counterparts [Rytsola et al., 2002; Saito et al., 2002]; signals from the PMTare digitized by means of ultrafast modules. Digitized pulses are stored in a personalcomputer and analyzed off-line by software; digital filters select pulses of suitableshape and amplitude. Improvement in timing resolution is obtained with respect to thestandard configuration of the positron lifetime setup, without decreasing the countingrate [Becvar, 2007]. Furthermore, it is possible to accumulate two independent lifetimespectra at a time by exploiting each detector as a source of start and stop signals.Introduction of ultrafast digitizers represents a real milestone in the PALS technique.

EXPERIMENTAL SETUP 397

The annihilation lifetime spectrum assumes the form of a histogram (generallycontaining from 1 to several million counts) which can be analyzed by means ofa computer program. The experimental spectrum is the convolution of the intrinsicspectrum with a resolution function, that is, a function describing the response ofthe apparatus to two simultaneous events. This can be obtained [Jean and Schrader,1988] from the time spectrum of 60Co, which decays by emitting two gamma rayswith similar energies (1.33 and 1.17 MeV) within a time interval of about 0.7 ps. Thetwo events can be considered simultaneous in the typical time scale of PALS, and thecorresponding time spectrum is often assumed to represent the resolution function ofthe positron annihilation lifetime spectrum, although the energy of the annihilationphoton is rather different with respect to the 60Co gamma rays. In fact, the full widthat half maximum (FWHM) for a time spectrum with 1.274- and 0.511-MeV gammasis larger (by a factor of about 1.1) than that for 1.17- and 1.33-MeV gammas.

A better approach is to use the time spectrum of 207Bi, which decays to an excitedstate of 207Pb with a half-life of 30 years. In the deexcitation process of 207Pb tothe ground state, two gammas, with respective energies of 1.06 and 0.57 MeV, areemitted, with a lifetime of 182 ps. The energies of the two photons are very near thosecorresponding to the start and stop photons in PALS. Deconvolution of the 207Bi timespectrum with a single fixed lifetime provides a realistic resolution function, whichmay be represented by a single Gaussian or a sum of Gaussians with different centroidsand weights; values between 150 and 300 ps are quite common for the FWHM of theresolution function.

Various computer codes are available [Kirkegaard and Eldrup, 1972; Gregory,1991; Shukla et al., 1993; Kansy, 1996] to analyze the annihilation lifetime spectrumin terms of different components, each corresponding to a particular positron state. APALS spectrum thus consists of the sum of a number of components, which can betreated as discrete or/and continuous. In the first case, each annihilation component isan exponential function of the form (I/τ) exp {−t/τ}, each characterized by a lifetimeτ and an intensity I. The intrinsic spectrum S(t) can be written

S(t) = R(t) ⊗[

N∑i=1

Ii

τi

exp (−t/τi) + B

](10.1)

where R(t) is the resolution function and B is the constant background, representingthe spurious coincidence events, to be subtracted during the fitting procedure. Thesymbol ⊗ stands for the convolution operation. PALS analyses in terms of threecomponents (i.e., N = 3) are quite common. Typical lifetimes of free positrons (i.e.,those that do not form Ps) in polymers are around 0.4 ns. p-Ps lifetimes in condensedmatter are usually below 0.15 ns; o-Ps shows the longest lifetimes, generally in therange 1 to 10 ns.

A continuous PALS spectral component is constructed as a continuous sum ofdiscrete components and is characterized by three parameters: the intensity and thefirst two moments of the distribution of lifetimes, that is, the centroid (mean lifetime)and the second moment (standard deviation from the mean lifetime).


A distribution of o-Ps lifetimes is expected in a polymeric material and will mirrorthe hole volume distribution present in the amorphous zones. The computer pro-grams Melt [Shukla et al., 1993] and Contin [Gregory, 1991] both analyze the timeannihilation spectrum only in terms of continuous components, without any guessas to the shape of the distributions. Conversely, the code Positronfit [Kirkegaard andEldrup, 1972] provides analyses only in term of discrete components. The program LT[Kansy, 1996] is also able to provide for eventual distributions of lifetimes (assuminga lognormal distribution) and can be used for a mixed analysis, in the sense that eachcomponent can be chosen to be discrete or continuous. Of course, the statistics of aspectrum must be substantially better if continuous components are to be resolvedaccurately. The structure of the program LT is described briefly below for analysis interms of discrete components. The analysis of continuous components is discussedthoroughly in Chapter 11.

LT works on two procedures of calculation: deconvolution of the experimental dataand a search for the model’s optimum parameters through a nonlinear least-squaresfitting procedure. The program requires the following inputs:

� The initial and final channels on which the analysis will be carried out; thesemust be fixed by the user.

� the shape of the resolution function R(t), approximated as a sum ofGaussian functions with different weights, FWHM, and centroids. Alternatively,an exponential-sided Gaussian function—a Gaussian convoluted with two expo-nential functions—can be used. This is useful for the simulation of tails in theresolution function:

FIGURE 10.3 Positron lifetime annihilation spectrum of polypropylene.

POSITRONIUM IN AMORPHOUS POLYMERS 399

� An approximate value for each lifetime, to start the analysis� A background value

It is possible to subtract a contribution due to annihilation events in the sourcesupport.

Any of the parameters mentioned above can be fixed by the user; otherwise, theprogram treats them as free parameters. The program outputs the lifetime and inten-sity of each component, the value of the background, and parameters characterizingthe resolution function; each free parameter is shown with the associated statisti-cal uncertainty. Furthermore, the result of a chi-squares test is given, quantifyingthe goodness of fit. In addition to the numerical outputs, the results can also be dis-played in graphical form, showing the experimental spectrum, the fitted spectrum, andtheir difference (residuals), together with each component. An example is shown inFigure 10.3. Residuals are useful since the presence of systematic oscillations is themarker of an unsatisfactory model.

10.3 POSITRONIUM IN AMORPHOUS POLYMERS

Positronium formation is a complex mechanism, not fully understood yet. Variousmodels have been proposed. The first is the Ore model [Goldanskii, 1986]: Ps forma-tion occurs when a positron captures an electron (e.g., from a molecule M) accordingto the following scheme:

e+ + M → Ps + M+ (10.2)

Reaction (10.2) is generally endothermic, due to the small binding energy Eb of Ps.Therefore, Ps formation is impossible below an energy threshold Et given by

Et = Ei − Eb (10.3)

where Ei is the ionization energy of the substance. There is also an upper limitEmax since at energies larger than those typical of electronic excitation and ioniza-tion, Ps is formed with a high kinetic energy and the species becomes unstable. Itis deduced that Ps formation occurs only when the positron energy falls within theinterval Et < E < Emax, called the Ore gap. This model works well in simple gases[Griffith and Heyland, 1978], but in molecular liquids and solids, other models providea better description of the experimental results. In the spur model [Mogensen, 1974]it is argued that a positron injected with rather high energy (on the average 200 keV,for the particles emitted by 22Na) slows down rapidly by ionization and excitationprocesses. Thus, in its terminal track (i.e., during the dissipation of the last 500 eV)the positron is surrounded by electrons and ionized and excited species. Ps forma-tion with one of the intraspur electrons may then occur in competition with geminaterecombination. This process can require up to several picoseconds, depending onthe medium, while in the Ore model, Ps formation is much faster. Although the two


models were long considered as alternatives, they should be viewed as complemen-tary, and various experiments have detected a fraction of Ps formed by each process[Bisi et al., 1983; Eldrup et al., 1983].

In recent years an extension of the spur model was proposed, the blob model[Stepanov et al., 2001, 2005; Stepanov and Byakov, 2002, 2003]. The authors remarkthat in the last part of the spur, about 30 electron–ion pairs are generated within aspherical volume (the blob) having a radius of about 4 nm. The positron can diffuseout of the blob and thermalize in a volume bigger than the blob. Electrons in the blobare prevented from diffusing by the strong electric fields of the positive ions and there-fore, to a first approximation the blob does not expand. Electron–ion recombinationprocesses decrease the number of electrons available in the blob with time. Some ofthe thermalized positrons outside the blob can still diffuse back to the blob and becometrapped inside it. According to this model, Ps formation is a multistage process. First,a positron encounters an electron by forming a weakly bound pair with an initialenergy (about 1 eV) which is lost by vibrational excitation. Subsequently, the pairmay escape from the blob, forming quasi-free Ps; finally, it becomes trapped in a sitewith reduced electron density, owing to the repulsive exchange interaction betweenthe Ps electron and the external electrons belonging to the surrounding molecules. Inthe case of polymers, such sites can be identified as free-volume holes.

The main difficulty encountered in testing the various models of Ps formationarises due to the fact that they involve fast processes (in the range of subpicoseconds)of radiation chemistry, in a medium (the terminal track of the positron) that is clearlynot in thermal equilibrium. In addition, the only parameter generated in an experimentis the o-Ps intensity, which depends in a complex way on many competing processes,which are very difficult to control.

In any event, independent of the model of Ps formation, the aforementioned trap-ping mechanism, based on the Pauli principle (exchange forces), explains well thecapability of Ps to probe free-volume sites in polymers. In this connection, PALS mea-surements in poly(ether ether ketone) (PEEK) [Nakanishi and Jean, 1989], preparedwith different degrees of crystallinity, have shown that o-Ps intensity is correlated lin-early with the percentage of amorphous phase of the sample. The extrapolated valueof the intensity at 100% crystallinity is zero, which strongly supports the idea that Psis trapped only in the free volume.

10.4 ESTIMATION OF HOLE SIZES FROM o-Ps LIFETIME

10.4.1 Tao–Eldrup Equation

A correlation between the annihilation decay rate and the size of the hole hostingPs was suggested by Tao [1972] and further elaborated by Eldrup et al. [1981]. Themodel assumes the hole to be a spherical void with an effective radius R. Such a Pstrap has a potential well of finite depth; however, for convenience of calculation, oneusually assumes the depth to be infinite, but increases the radius to R + ∆R, ∆R

being an empirical parameter [Nakanishi et al., 1988] that describes the penetration

ESTIMATION OF HOLE SIZES FROM o-Ps LIFETIME 401

of the Ps wavefunction into the bulk. The electron density is assumed to be zero forr < R and constant for r > R. The probability of finding ground-state Ps inside thebulk polymer is

p = 4π

∫ ∞

R+∆R

|ψ (r)|2r2dr = ∆R

R + ∆R+ 1

2πsin

2πR

R + ∆R(10.4)

where, as reported in quantum mechanics textbooks [Gol’dman and Krivchenkov,1993],

ψ(r) = sin[πr/(R + ∆R)]

r√

2π(R + ∆R)(10.5)

inside the well and zero outside. It is assumed that the annihilation rate of o-Ps in thebulk state is λ0 � 2 ns−1, which results from the spin-averaged annihilation rate ofp-Ps (8 ns−1) and o-Ps (0.007 ns−1) in a vacuum. The relationship between the o-Pspickoff decay rate λp (ns−1) and the hole radius R is

λp = λ0

[∆R

R + ∆R+ 1

2πsin

(2π

R

R + ∆R

)](10.6)

The o-Ps lifetime τ3 is the reciprocal of the total decay rate λ3, which is the sumof the pickoff decay rate and the intrinsic decay rate λi :

τ3 = 1

λ3= 1

λp + λi

(10.7)

The λi contribution, expressed as λi = ηPs/142, is included in Eq. (10.7) for thesake of completeness but is generally negligible when treating o-Ps lifetimes in poly-mers. The relative contact density ηPs = |ψ(0)|2/|ψ(0)|2vacuum is a phenomenologicalparameter that takes into account any perturbation of the Ps wavefunction ψ due tothe environment [Dupasquier, 1983]. Although geometries other than the sphericalone can be considered and produce different analytical relationships between λp andthe size of the hole, Eq. (10.6) is almost universally adopted to deduce the averagevolume of the holes as a function of external variables. However, it should not beforgotten that Eq. (10.6) is in no way a real physical law, since it is obtained underthe approximations sketched above. In any event, the fact that the shape of the holesis generally not spherical makes it appropriate to ask how much realistic it is to useEq. (10.6), to deduce: (1) the absolute sizes of the holes and (2) the variation infree-volume fraction versus a physical variable such as temperature.

10.4.2 o-Ps Lifetime versus the Cavity Size in Different Geometries

The Schrodinger equation, which yields the ground-state o-Ps wavefunction, can besolved analytically for potential wells that are different from spherical. The sameassumptions as those used to derive Eq. (10.6) produce the following formulas:


Cubic cavity [Jasinska et al., 1996, 1999]:

λp = λ0

[1 −

(a

a + 2∆R+ 1

πsin

πa

a + 2∆R

)3]

(10.8)

where a is the length of the cube edge.Prism with square cross section [Jasinska et al., 1996, 1999]:

λp = λ0

[1 −

(a

a + 2∆R+ 1

πsin

πa

a + 2∆R

)2

×(

ma

ma + 2∆R+ 1

πsin

πma

ma + 2∆R

)](10.9)

where m is the axial length and a is the transverse length. The expression for a cubeis recovered with m = 1.

Layered structure [Jasinska et al., 1996, 1999]:

λp = λ0

[1 −

(d

d + 2∆R+ 1

πsin

πd

d + 2∆R

)](10.10)

where d is the distance between the walls of the cavity.

Infinite cylindrical channel with radius R [Jasinska et al., 1996, 1999]:

λp = −2.56∫ a1[R/(R+∆R)]

a1

J20 (r)r dr (10.11)

where J0(x) is the zero-order Bessel function of the first kind and a1 = 2.4048 is itsfirst zero.

Finite cylinder with radius R and height u [Olson et al., 2002]:

λp = λ0 (1 − P1P2) (10.12)

where

P1 = u

u + ∆u+ 1

πsin

πu

u + ∆u(10.13a)

P2 =∫ a1R/(R+∆R)

0 xJ20 (x) dx∫ a1

0 xJ20 (x) dx

(10.13b)

ESTIMATION OF FREE-VOLUME FRACTION 403

It can be observed that an infinite channel with a square cross section is obtainedfrom Eq. (10.9) when m → ∞. Prisms and layered cavities have been used, forexample, in structural studies of clays [Joshi et al., 1998; Consolati et al., 2002].A discussion of pore-size distributions in low-dielectric thin films [Gidley et al.,2000] was based on cubic structures. Of course, other geometries are possible: Forexample, ellipsoidal holes were also considered [Jean and Shi, 1994], in an interestingattempt to frame free-volume holes in semicrystalline polymers subjected to tensiledeformation.

10.5 ESTIMATION OF FREE-VOLUME FRACTION

In Section 10.4 we discussed how to deduce from o-Ps lifetimes the sizes ofholes hosting Ps, which depend only slightly on the geometry adopted for a giveno-Ps lifetime [Consolati, 2002]. However, to determine the absolute free-volume frac-tion f it is necessary to couple PALS data with the results from another technique.Specific volume measurements are particularly suitable. It is possible to obtain auseful expression for f by combining PALS and dilatometry results [Srithawatponget al., 1999]; the specific volume V is written as the sum of the specific occupiedvolume Vo and the free-volume contribution, as follows:

V = N ′vh + Vo (10.14)

where N′ is the free volume number density per unit mass and vh is their mean volume.Since both V and vh are known as a function of temperature, it is possible to plot Vdirectly versus vh, which yields values for N′ and Vo by means of a fitting procedure.With these results, the free-volume fraction can be calculated:

f = N ′vh

V(10.15)

An alternative approach is to make use of the o-Ps intensity I3. This parameter hasbeen proposed [Wang et al., 1990] to be linearly correlated to the number density offree-volume holes, so that the free-volume fraction can be written only in terms ofparameters derived from PALS:

f = C′vhI3 (10.16)

where C′ is a constant which, in principle, may depend on polymer structure. Thevalue of C′ can be obtained by comparison with values of f derived from specificvolume data via appropriate theory. Equation (10.16) seems appealing for its sim-plicity and reasonableness, however, it should be considered cautiously since it is apurely phenomenological relationship; it has been criticized [Shantarovich, 1996;Wang et al., 1998] since several factors (due primarily to intraspur or intrablobreactions) contribute simultaneously to Ps formation, and it is generally impossi-ble to isolate from among them the influence of the number density of holes. In


particular, sequential PALS measurements of PMMA samples [Wang et al., 1998]over a wide range of temperatures showed that o-Ps intensity displays hysteresis; theeffect has been attributed to processes of radiation chemistry in the terminal track of thepositron, and in this case I3 does not reflect solely the variation in the number densityof holes with temperature. On the other hand, refreshing the sample [Srithawatponget al., 1999] by heating above the glass transition temperature before each measure-ment in the glass seems to erase, at least partially, spur effects on I3, which appearless important when the polymer is above Tg . Phenomenology appears to be ratherintriguing and worth analyzing more thoroughly by further experiments. In the subse-quent discussion we use Eqs. (10.14) and (10.15) to get the free-volume fraction fromPALS data.

Returning to dilatometry, such data are essential not only to obtain the experimentalfree volume together with PALS data, but also to evaluate the holes fraction h withinthe framework of the lattice–hole theory [Simha and Somcynsky, 1969]. Concerningthe last point, the theory supplies an equation of state that is valid for any polymericstructure; at atmospheric pressure it is expressed in terms of reduced thermodynamiccoordinates T = T/T ∗and V = V/V ∗, where T* and V* are characteristic parametersthat depend on the specific material. The parameters T* and V* for the materialinvestigated result from a fitting procedure; to this end, the following expression[Simha et al., 1973] produces a quite accurate fit at atmospheric pressure and has beenused successfully for a wide class of polymers [Simha and Wilson, 1973; Rodgers,1993], including oligomers [Wilson and Simha, 1973]:

lnV

V ∗ = A + B

(T

T ∗

)3/2

(10.17)

where T is the absolute temperature; the values of the coefficients A and B are:−0.1035 ± 0.0003 and 23.85 ± 0.03, respectively [Utracki and Simha, 2001]. WithV* and T* extracted from experiment, the hole fraction in the melt, again at atmo-spheric pressure, is given by the expression [Simha and Wilson, 1973]

h = V − K (T/T ∗) V ∗

V(10.18)

K is a slowly varying temperature function that can be assigned a value of 0.954within a limited range of temperatures above the glass transition. Equation (10.18)clearly illustrates the nature of the hole fraction as a free-volume fraction.

In the following we discuss various experimental data, with the aim of show-ing that use of nonspherical geometries for holes can in some cases supply betteragreement between free-volume fractions f and h than can the spherical model. Fur-thermore, results in some polymers suggest a nonisotropic growth of the cavities withtemperature.

DISCUSSION OF SOME EXPERIMENTAL RESULTS 405

10.6 DISCUSSION OF SOME EXPERIMENTAL RESULTS

10.6.1 Polystyrene

Yu et al. [1994] carried out PALS measurements on four PS fractions (4, 9, 25,and 400 kDa, respectively) versus temperature (Figure 10.4). They evaluated thefree-volume fractions on the basis of the proportionality between the free-volumefraction as probed by o-Ps and the product of the o-Ps intensity I3 and the meancavity volume assumed spherical, as sketched previously [Eq. (10.16)]. On this basisthey observed agreement with the free-volume fraction predicted as given by thelattice–hole model [Simha and Somcynsky, 1969] over a range of temperatures aboveTg , the proportionality constant C′ being a molar mass–dependent fitting parameter.

Later, Olson [2003] revisited these data using the procedure shown in Eqs. (10.14)and (10.15). Whereas the previous analysis had yielded apparent agreement betweenthe temperature-dependent PALS free volume and thermodynamic hole fraction, thelatest analysis indicated systematic departures. Figure 10.5 shows the correlationbetween specific volume and average hole volume, evaluated at the same temperature;the linearity of the four plots demonstrates the validity of Eq. (10.16) with a constantvalue of N′, the number density of holes. Incidentally, we note that the constancyof I3 above Tg found by Yu et al. [1994] is consistent with the constant value ofN′, if I3 is assumed proportional to the hole density. In the analysis a constant valueof the occupied volume was assumed; nevertheless, we checked [Consolati et al.,2005], using the detailed data [Simha and Wilson, 1973], that the assumption ofa temperature-dependent occupied volume does not affect the estimate of N′, thevariation being on the order of 5%. Since dilatometric measurements were not reported[Yu et al., 1994], specific volume data were evaluated using the empirical expressions

FIGURE 10.4 o-Ps lifetime in polystyrene fractions. (From Yu et al. [1994].)


FIGURE 10.5 Specific volume V versus the average hole volume vh (in spherical approxi-mation) in the PS fractions.

of Uberreiter and Kanig [1951] for the molar mass dependence of V. A comparisonwith a polystyrene of molar mass 9000 listed by Zoller and Walsh [1995] shows onlysmall differences; the hole fraction h at the glass transition temperature calculatedfrom these data differs from that evaluated according to Uberreiter and Kanig by lessthan 2.2%, and differences in the slopes are below 1.6%.

Determination of N′ and Vo from PALS and specific volume data [Eq. (10.14)]allows one to estimate the free-volume fraction f [Eq. (10.15)], which is displayedin Figure 10.6 for all the molar masses investigated, together with the correspondingtheoretical fractions h. We note a systematic discrepancy between the two functionsin the sense that f is larger than h, in particular for the smaller molar masses.

To check if these differences can be ascribed to the choice of hole shape, weexplored [Consolati et al., 2005] two different geometries, cylindrical and prismaticholes. Transformations from o-Ps lifetimes to the radius and edge of cylindrical andprismatic holes, respectively, were performed according to Eqs. (10.12–10.13) and(10.9), respectively. Parameters q (=u/R) and m [Eq. (10.9)] are the aspect ratios for acylinder and a prism with a square section, respectively. We calculated the free-volumefractions f, using m and q as fitting parameters, to produce the best agreement with thetheoretical fraction h. The results are displayed in Figure 10.6, and the correspondingvalues of m and q are shown in Table 10.1, together with the associated values ofN′ and Vo for all three hole geometries. From Table 10.1 we see that agreement isobtained for aspect ratios less than unity. Relative errors in q and m are evaluatedto be around 20%; they are based on conservative estimates and do not arise onlyfrom the scattering of experimental results. We note the close similarity of the twosets of asymmetries. Our results suggest that holes can be treated equivalently as


FIGURE 10.6 Free-volume fractions in PS: theoretical (continuous lines) and estimatedfrom PALS assuming cylindrical (open triangles), prismatic (filled squares), and spherical(filled circles) holes.

flattened disks or prisms; we also explored the possibility of values of m and q > 1,corresponding to rodlike and barlike holes, but did not obtain satisfactory fits.

From Table 10.1 we observe that the number density of holes, N ′, depends on thegeometry, since it changes by a numerical factor on passing from spheres to disks toprisms; therefore, caution is necessary when discussing this quantity, since only theorder of magnitude can be meaningful. The amount of free volume N ′vh does dependon the shape assumed for the cavity, as is evident by comparing the data in Figure10.6. In the PS fractions investigated, prismatic and cylindrical holes produce thesame amount of free volume, which is lower than the free volume for spherical holes.

We deduce that the same o-Ps lifetime produces different free-volume fractions,according to the various models adopted. Of course, real holes are shaped irregularly,and a simple geometry can be a rather crude approximation; nevertheless, comparisonbetween PALS data and theoretical free-volume predictions allows one to gain furtherinsight into hole features by providing some knowledge of their morphology.


TABLE 10.1 Number Density of Holes N′, Occupied Volume Vo, and AspectRatios q, and m Estimated by PALS in Four PS Fractions According to DifferentHoles Shapes

Spherical Holes Cylindrical Holes Prismatic Holes

N ′ Vo N ′ Vo N ′ Vo

M.W. (Da) (1021 g−1) (cm3/g) (1021 g−1) (cm3/g) q (1021 g−1) (cm3/g) m

4,000 0.614 0.886 0.327 0.899 0.25 0.149 0.899 0.289,200 0.650 0.885 0.333 0.900 0.25 0.114 0.899 0.23

25,000 0.603 0.898 0.800 0.900 0.54 0.358 0.900 0.63400,000 0.621 0.897 0.489 0.900 0.31 0.223 0.900 0.35

10.6.2 Oligomeric Series

Puzzling results were found for two oligomeric series [Consolati, 2005, 2006]. Fiveoligomeric perfluoropolyethers (Fomblin-Zdol from Ausimont S.p.A.) with the struc-ture

HO CH2CF2 O (CF2CF2 O )a (CF2 O)b CF2CH2 OH

were investigated. Molecular weights and glass transition temperatures Tg are reportedin Table (10.2). Dilatometric measurements were carried out [Messa, 1999] over atemperature range encompassing Tg ; the specific volume V shows a linear relationwith temperature in the liquid phase as well as in the glassy phase, with differentslopes (Figure 10.7).

Figure 10.8 shows the o-Ps lifetime τ3 plotted versus the temperature ratio T/Tg.All the structures investigated show the same behavior: a linear trend below Tg,followed by a dependence on the temperature that is still linear but with a higherslope than below Tg. Finally, at the highest temperatures, the increase in τ3 becomesmore gradual.

Such behavior partly mirrors that of the free volume; indeed, below Tg, inter-chain motions are strongly hindered and the free volume shows only small variations

TABLE 10.2 Molecular Weights, Glass Transition Temperature Tg, CharacteristicReducing Parameters V∗ and T∗, Exponent k, Number Density of Holes N′, andExperimental (Vo) and Theoretical (V th

o ) Occupied Volume for the PerfluoropolyethersInvestigated

M.W. (Da) Tg (K) V ∗(cm3/g) T ∗(K) k N′ × 1021 (g−1) Vo (cm3/g) V tho (cm3/g)

439 189 0.531 6954 2.13 2.6 0.508 0.506947 171 0.502 6396 2.15 2.2 0.479 0.479

3250 157 0.480 6143 2.25 1.9 0.456 0.4574610 153 0.478 5892 2.40 1.4 0.451 0.4569410 153 0.474 6062 2.40 1.4 0.448 0.452


FIGURE 10.7 Specific volume versus temperature in oligomeric perfluoropolyethers. (FromMessa [1999].)

with temperature, and above Tg, chain segments become mobile and the free vol-ume increases with the temperature at a higher rate, so the expansion coefficient ishigher than in the glassy state. The range of highest temperatures, where dτ3/dT

again becomes moderate, was not considered for evaluation of the free-volume

FIGURE 10.8 o-Ps lifetime τ3 as a function of the temperature ratio T/Tg in the perfluoro-polyethers investigated.


fraction since there is general agreement that o-Ps in this region does not providecorrect information on the free volume. Various explanations of this high-temperaturebehavior have been suggested: Formation of a Ps bubble in the liquid phase [Mikhinet al., 2003], creation of holes by Ps itself [Ito et al., 1996], or the relaxation timeof the molecular chains becomes comparable to τ3 [Bartos et al., 2001]. This lastinterpretation is supported by dielectric spectroscopy measurements. In each of thesethree situations, Ps will not be able to probe correctly the sizes of the voids in whichit is trapped.

Transformation from o-Ps lifetime to hole volume was first carried out using thespherical approximation [Eq. (10.6)]. Next, the specific volume V for the structuresinvestigated (evaluated at the same temperatures as the positron data) was plottedversus the hole volumes vh (Figure 10.9), and as in the case of the PS fractions, thedata are fitted by straight lines, with correlation coefficients higher than 0.99. Againthis means that the number density of holes N′ can once more be assumed constant(i.e., independent of the temperature) for T > Tg . It was verified that the result doesnot depend on the approximation adopted: Indeed, cubic, cylindrical, or prismaticgeometries produce the same linear correlation, although with slightly different valuesfor N′ and Vo .

Fractional free volume f was then calculated according to the previous definition[Eq. (10.15)]. Its variation with temperature is shown in Figure 10.10, together withthe theoretical free-volume fraction h; values of f (obtained assuming spherical holes,plotted as circles in Figure 10.10) are systematically lower than h for all the structures.Furthermore, the expansion coefficients of f are higher than the corresponding valuesdeduced from the theory.

FIGURE 10.9 Specific volume V as a function of hole volumes vh (calculated in sphericalapproximation) in oligomeric perfluoropolyethers.


FIGURE 10.10 Free-volume fractions evaluated from PALS and dilatometric data (f) andtheoretical (h) (continuous lines). Circles, spherical holes; squares, anisotropic holes.

It is worth pointing out that this result does not depend on the spherical shapeadopted for the estimate of the hole size; indeed, essentially identical discrepan-cies with the theory are obtained using cubic, prismatic, or cylindrical holes. Allthese models assume an isotropic expansion of holes; that is, vh ∝ s3, where s is thecharacteristic dimension of the hole hosting Ps. The values of s, as evaluated fromany model, are necessarily approximate estimates; the irregular shape of real holesprecludes the deduction of an exact value of the cavity size. Equations (10.6), and(10.8)–(10.13) are all analogous in that they obtain from τ3 a characteristic hole dimen-sion s, whose numerical value depends only slightly on the model adopted. This is apossible reason that discrepancy between f and h persists whatever shape is assumedfor holes.

Thus, assumptions of anisotropic hole expansion yield a different relationshipbetween hole volume and the characteristic dimension(s) s, which are allowed toincrease. For example, by framing the holes as cylinders with a fixed height σ, theirvolumes will increase according to the square of the radius s: vh ∝ σs2. As a further,naive example, if the cavities are modeled as voids between two parallel planes (e.g.,the interlayer gaps in clays [Consolati et al., 2002]), the increase in their volume


should be linearly related to the distance between the planes. In such cases, thethermal expansion coefficient of the free volume should be lower than in the case ofspherical holes. More generally, since macromolecular motions are partially hinderedby physical or chemical constraints such as entanglements, cross-links, or hydrogenbonds, it can be supposed that the volume expansion of voids follows a power law withrespect to the dimension s, which changes while the other, constrained dimensions σ

remain fixed:

vh ∝ σ3−ksk (10.19)

Isotropic expansion (i.e., unconstrained holes) implies that k = 3. For the per-fluoropolyethers investigated, we anticipate that agreement with the theory can beexpected for k < 3. To obtain k, the o-Ps lifetime τ3 must be expressed in terms of theaverage constrained and unconstrained dimensions of the cavity.

A simple model for holes framed within a cylindrical geometry was employed forthis purpose; we used Eqs. (10.12)–(10.13) for a cylinder, but the expansion along thedirection of the radius s was assumed to be free, whereas the height u was assumedto grow slower than the radius:

u

u0=

(s

s0

)c

(10.20)

where u0 and s0 are the values corresponding to the lowest temperature investi-gated and the exponent c is expected to be smaller than 1 (c = 1 corresponds toisotropic expansion). To make a quantitative estimate, the aspect ratio r = u0/s0 = 3was introduced. The following expression for hole volume results:

vh = πus2 = πu0s2+c

sc0= πrs1−c

0 s2+c = πrs3−k0 sk (10.21)

where k = 2 + c in the last identity. Equation (10.21) has the same form as Eq. (10.19)and was used to calculate the hole volume for a fixed k. Fitting the specific volumeV versus vh by straight lines (analogous to that used for spherical holes) yielded theparameters N′ and Vo for each k. These quantities were then substituted into Eq.(10.15) to compute the fractional free volume f, which was subsequently comparedto h.

Values of the exponent k giving the best agreement between f and h (Figure 10.10,squares) are shown in Table 10.2 for each molar mass; the values range between 2and 3 and are rather similar to each other, the average value being 2.3 ± 0.1. Similarvalues of k were found using different aspect ratios r, the differences in the values of kbeing on the same order (10%) as the statistical uncertainty associated to it. Framingthe holes as anisotropic prisms produced almost identical values of k. In Table 10.2the values of N′ and Vo are also shown; the number density of holes tends to decreasewith increasing the molecular weight, in agreement with expectation, since the higherthe molecular weight, the lower the density of chain ends that contribute to the free


volume. Of course, only the order of magnitude of N′ should be considered, since theproportionality factor between vh and sk depends on the choice of r and s0. Occupiedvolumes Vo determined experimentally from Eq. (10.14) compare favorably with thevalues V th

o calculated from the lattice–hole theory (the last column of Table 10.2), thedeviations being below 1.2%.

Similar results were obtained for an homologous series of polypropylene glycols(PPGs) [Consolati, 2006]. Five members with molecular weights 76 (monomer), 134(dimer), 192 (trimer), 1000, and 4000 Da were investigated. Also in this case, the spe-cific volume V shows [Messa, 1999] linear trends with temperature T (Figure 10.11);in the range T > Tg , data are well approximated by straight lines. The variation of theo-Ps lifetime τ3 with the temperature ratio T/Tg is shown in Figure 10.12 and is quitesimilar to that of the perfluoropolyethers. A linear correlation was found betweenV and vh (assuming spherical geometry) for all the PPGs investigated and yieldedvalues of the parameters N′ and Vo.

The fractional free volume f was evaluated at the various temperatures and com-pared to h. The results, displayed in Figure 10.13, show a systematic deviation forall the PPGs investigated in the case of isotropic hole expansion, as for the perfluo-ropolyethers examined above, although the difference is reduced for the monomer.The discrepancy does not depend on the adopted geometry and is again eliminatedby assuming anisotropic expansion for holes [Eq. (10.21)] and treating k as a fittingparameter. The values of k supplying the best agreement between f and h (Figure 10.13,full squares) are around 2, except for the monomer; they are reported in Table 10.3.The last two columns of Table 10.3 show the results of a χ2 test between h and f. Thislast was evaluated assuming isotropic (χ2

isotr) and anisotropic (χ2anis) holes, respec-

tively. It is evident from Table 10.3 that the isotropic assumption is acceptable onlyfor the monomer.

FIGURE 10.11 Specific volume as a function of temperature in PPG. (From Messa [1999].)


FIGURE 10.12 o-Ps lifetime τ3 versus the temperature T/Tg in the PPG investigated.

FIGURE 10.13 Theoretical free-volume fractions in poly(propylene glycol) (solid lines) hare compared with f by assuming spherical (circles), cylindrical (triangles), or cubic (opensquares) holes. Filled squares, holes expanding anisotropically.

CONCLUSIONS 415

TABLE 10.3 Molecular Weights, Glass Transition Temperature Tg, CharacteristicReducing Parameters V∗ and T∗, Exponent k for the PPG investigated, and χ2 TestResulting from a Comparison Between the f and h Free-Volume Fractions withIsotropic (χ2

isotr) and Anisotropic (χ2anis) Holes

M.W. (Da) Tg (K) V ∗ (cm3/g) T ∗ (K) k χ2isotr χ2

anis

76 189 0.919 8768 2.7 1.9 0.57134 196 0.910 7668 1.7 20.3 0.04192 193 0.911 7611 1.9 19.0 0.11

1000 199 0.919 7419 2.2 9.6 0.044000 199 0.922 7486 2.0 15.3 0.02

The findings above support the idea of an anisotropic expansion of holes with thetemperature in PPGs, at least for molecular weights of dimer and above. Only inthe absence of constraints (an ideal case) or with constraints acting with the sameintensity in all directions, does one expect to find an isotropic expansion. Such asituation could explain the result for the monomer, whose molecules are arrangedmore symmetrically with respect to the other PPG samples. Indeed, in this case thevalue of k deduced from the fit to experiment is the highest and quite near 3.

10.7 CONCLUSIONS

In conclusion, PALS in combination with specific volume measurements makes pos-sible the determination of the variation of the free-volume fraction versus temperature.The result depends on the geometry adopted for the holes as well as on whether thehole expansion is isotropic or anisotropic. In the case of isotropic expansion, numeri-cally different free-volume fractions can be obtained, according to the various shapesadopted for holes. Comparison between PALS data and theoretical predictions there-fore allows one to gain further insight into hole structure by providing information ontheir morphology; indeed, in the case of PS a better agreement with the predicted freevolume is obtained when holes are framed as disks or prisms rather than spheres. Thisdoes not, of course, imply that the spherical geometry necessarily produces flawedresults.

In the case of the two series of oligomers discussed above, excellent agreementis obtained between PALS results and the theoretical free-volume fraction if theassumption of isotropic expansion of the cavities with temperature is removed. Theexponent k of the power law relating unconstrained average size and the volume ofthe holes is less than 3. Of course, the model of anisotropic growth is rather crudesince it neglects the possibility that the constraints can change with the dimensions ofthe holes; in this connection, k could then be expected to be temperature dependent.Nevertheless, the model is compatible with the dynamics of macromolecules whosemotions are generally subjected to constraints.

Future developments could consider distributions of non-spherical holes, with thepossibility of changing shape on growing from smaller to larger holes, as well as


distributions of anisotropic holes, with the exponent k now dependent on the size ofthe hole. PALS analyses should be compared not only to the analytical theory but alsoto realistic simulations, due to the higher level of complexity that can be incorporated.Extension to the glass of the analytical theory (which, strictly speaking, is valid inthe melt) will also support PALS results in this challenging phase.

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Polymer Physics (From Suspensions to Nanocomposites and Beyond) || Morphology of Free-Volume Holes in Amorphous Polymers by Means of Positron Annihilation Lifetime Spectroscopy