The focus of this article is the application of neutronscattering in polymer science, with various examples givento illustrate the types of problems addressed. Neutronscattering is a powerful analytical tool for investigatingpolymers. Unique aspects such as the capacity to measurethick samples with an absence heating or radiation-induceddamage, selective control of the scattering contrast byisotopic substitution, and the measurement of dynamicproperties as a function of length scale are among themany attributes that make neutron methods not justcomplementary to other analytical techniques, but in manycases the only means to address material issues. Theinformation that can be obtained from neutron scattering –the size, form, and orientation of molecular chains,
the thermodynamics and phase structure of mixtures,interfaces and interfacial phenomena, the dynamics at boththe local and global length scales, details of the structure ofcomplex nanocomposites and biological materials, etc. –has only been surveyed herein. It is a testament to the utilityof neutron scattering that even though the application ofneutrons to polymers did not begin in earnest until the1990s, through the past decade, about 325 peer-reviewedpublications per year have appeared on the topic. Whetherthis figure changes in the future will likely depend on theavailability of the specialized, costly facilities required forneutron experiments. Certainly there is no expectation ofabatement in the number of important problems in polymerscience that can be addressed using neutron methods.
1 INTRODUCTION
Neutrons are subatomic particles that stabilize the nucleiof atoms against the electromagnetic repulsions betweenprotons. Impingement on atoms changes the direction ofneutrons, and the measurement of the scattering patternreveals the relative positions of the atoms. Neutrons canalso exchange energy with target nuclei, and analysis ofthis exchange provides information on motions of theatoms. Thus, neutron scattering probes both the structureand dynamics of materials, making the experiments aunique scientific and technological resource. Advantagesof neutron scattering experiments include
• Neutrons lack an electric charge and are nonionizing.Their weak interaction with matter makes them verypenetrating, and thermal neutrons cause negligibleradiation damage. In combination with a lack ofsample heating, this makes neutrons well suited forinvestigation of thick (macroscopic) samples. Also,many materials used in the construction of cryostats,sample cells, etc. are transparent to neutrons.
• Small-angle neutron scattering (SANS) uses neutronshaving wavelengths of the order of 10−10 m (5–20 A),which correspond to thermal neutrons with energiesin the 5 to 100 meV range (≈ kBT , where kB is theBoltzmann constant). The wavelength is comparableto interatomic spacing, so that scattering at smallangles reveals structural information on length scalesof interest to chemists. The broad dynamic rangeallows the study of molecular rotations, vibrations,and translations and is especially useful for polymers,whose motions span many decades of frequency.
• The interaction (absorption and scattering) ofneutrons by elements varies in no systematic fashionacross the periodic table, and likewise among isotopes(e.g. hydrogen and deuterium); thus, labeling and
selective probing are extremely useful with SANSthrough control of the scattering contrast.
• Neutron scattering is sufficiently weak that multiplescattering effects are avoided. And unlike IR andRaman spectroscopies, there are no selection rulesrestricting the allowed transitions in neutron scat-tering.
Disadvantages are:
• SANS instrumentation requires dedicated facilities.The neutron sources are expensive to build, operate,and maintain. Small-angle X-ray scattering (SAXS)instruments, in comparison, are relatively low costand common in individual laboratories.
• The intensity (neutron flux) is lower than that forX-rays, and consequently measurement times tend tobe long, and it often requires large amounts of sample(which can be a particular problem for biologicalinvestigations).
• The incoherent scattering cross-section for 1H is largeand can contribute an unwanted background signalthat interferes with analysis of coherent SANS data.
Neutron and X-ray scattering for structural studiescover comparable spatial ranges; however, there areimportant differences between the methods. While X-raysinteract with the electron cloud external to the nucleus,neutrons are scattered by the nucleus (or for polarizedneutrons by the magnetic moment of unpaired electrons).The strength of the neutron–nucleus interaction isquantified by the scattering length, b, a property that, asstated, varies randomly across the periodic table and candiffer between isotopes of the same element. In particular,deuteration of a compound can effect large changes incontrast, due to the substantial difference in scatteringlength between hydrogen, b = −3.739 × 10−13 cm, anddeuterium, b = +6.671 × 10−13 cm. The contrast in aSANS measurement depends on the squared differencein scattering lengths (Equation 3), which means that large
differences in scattering can be achieved between portionsof a molecule or among different components of a sample.This property is exploited in many neutron studies.
Neutrons for SANS are produced either as a continuousflux from a nuclear reactor or pulsed, usually from a spal-lation source. A moderator is used to slow the neutronsto thermal velocities (<10 km s−1), enhancing the resolu-tion. The reactor output is filtered to monochromatize andremove γ-rays, and the beam is also collimated. Spalla-tion sources use a particle accelerator to produce a beam,typically protons, which displaces neutrons from a targetof heavy nuclei (183W or 238U). Higher neutron fluxes areachieved, and even when operating in a pulsed mode, thecontinuous neutron output is about two orders of magni-tude higher than that from reactors. The newer neutronsources tend to use particle accelerators. Detectors forneutron scattering are usually two-dimensional (2-D) areadetectors. 3He gas is the common active medium, with theincident neutron causing a reaction that forms a protonand a triton. These charged particles are attracted to ananode, with the detected electrical signal proportional tothe charge originating with the neutron/3He reaction.
Although neutron scattering experiments requireaccess to a neutron research facility (there are about 30worldwide), a vast literature has accumulated describingneutron scattering studies of polymers, colloids, metals,glasses, ceramics, biological materials, magnetic solids,superconductors, etc. Herein, the focus is the applicationof neutron scattering in polymer science, with examplesgiven to illustrate the types of problems addressed;suggestions for further reading are provided at the end.
2 BASIC CONCEPTS OF NEUTRONSCATTERING
The basic neutron scattering experiment is shown inFigure 1. Neutrons from the incident beam interact withthe sample and are either blocked by a beam stop
Incident neutronBeam
Sample Beamstop Detector
Scattered beam
θ
Figure 1 Depiction of typical elastic SANS experiment.
or scattered onto the detector. The scattered neutronscan change both their energy, E, and direction, thelatter expressed as the modulus of the scattering vector,|Q| = 4π/λ sin θ/2, where λ is the neutron wavelengthand θ the scattering angle. Q (also designated as q, h,k, and s) is sometimes referred to as the momentumtransfer, because it is proportional to the change inmomentum of the scattered neutron. Elastic scatteringimplies no energy change (�E = 0), and thus onlyspatial information is provided on the scattering system.Inelastic scattering involves energy exchange due tomotion of the scattering nuclei; these measurementscan yield information about the dynamics. Commonly,inelastic scattering experiments are termed ‘quasielastic’,in reference to the small energy change compared withthe incident energy. Neutron energies for scatteringexperiments are a few meV, several orders of magnitudeless than the energy of an X-ray photon. The neutronenergy is also small compared with the energies associatedwith vibrational motion, with vibrational scattering peakspresent in the spectra approximately equidistance fromthe elastic peak. Energy differences for rotational andtranslational motion are small compared with the neutronenergy, and thus exchange is manifested as a broadeningof the elastic peak.
Two less common applications of neutron scattering areultra-small-angle neutron scattering (USANS) and wide-angle neutron scattering (WANS). While conventionalSANS experiments rely on pinhole collimation and abeam stop to obtain angular resolution, USANS usesa Bonse–Hart camera,(1) in which the incident andscattered beams are reflected multiple times throughsilicon crystals, precision-cut to maintain parallelism ofthe refracting planes.(2 – 4) The minimum angle accessiblewith USANS is about two decades smaller than that in aSANS experiment, thus overlapping with light scatteringmethods. For WANS, Q can extend to tens of A, coveringatomic distances. The method is analogous to wide-angle X-ray diffraction measurements; however, isotopicsubstitution can be employed with WANS to obtain,for example, radial distribution functions for chemicallydistinct hydrogens in a substance(5) or to circumventselection rules by breaking the symmetry of the chemicalstructure.(6)
A measure of the scattering power of a sample isthe differential scattering cross-section, d�/d�, whichis the ratio of incident neutrons scatterer per unit timeinto the solid angle ��. This cross-section is the sumof an angle-independent incoherent component, due toscattering from uncorrelated nuclei, and the coherentscattering. The former is used for inelastic measurementsprobing molecular motions; however, only the coherentscattering contains the phase information necessary toprovide structural information (size and shape of the
scattering entities). Integration over all energies andangles gives the total scattering cross-section, which isthe sum of the probabilities for neutron scattering withany energy change at all angles. Counting only the numberof scattered neutrons regardless of energy,
d�
d�=
∫ ∞
0
d2�
d�dEdE (1)
The measured intensity in the absence of multiplescattering is
I (Q) = I0��εdetV Td�(Q)
d�(2)
where I0 is the incident neutron flux, εdet the detectorefficiency, and V and T are the sample volume andtransmission. V , the product of the sample length andthe area of the incident beam, is typically in the rangefrom 0.01 to 5 cm3. When the scattering is dominatedby the coherent intensity, T must be large (>99%)to avoid multiple scattering, which would distort themeasured angle dependence. However, if the incidentbeam is attenuated due to strong incoherent scatteringor absorption, than the optimal coherent intensity isachieved for T = 37%. Standard scatterers can be usedto convert the measured data to absolute intensities.
This coherent cross-section is related to properties ofthe material according to
d�(Q)
d�
∣∣∣∣coh
= φ�ρ2VPP(Q)S(Q) (3)
where φ is the volume fraction of scatterers within thesample, each having a volume Vp. The squared differencebetween scattering length densities, �ρ2 = (ρ2 − ρ1)
2, fortwo regions or phases in a sample is referred to as thecontrast. The scattering length density for a molecule isthe sum over all atoms ρ = ∑
biρiNA/Mw, where ρi is themass density, NA is Avogadro’s number, and Mw is themolecular weight.
Listed in Table 1 are scattering length densities forcommon polymers and solvents. As noted earlier, adistinctive feature of neutron scattering is the contrastbetween different elements and isotopes, so that thecontrast can be adjusted by deuteration or judiciousselection of a solvent. However, because the C-Dbond distance is a few tenths of a percent smallerthan that for C-H bonds (due to a smaller vibrationalamplitude), the molecular polarizability is altered byisotopic substitution. This changes the interaction energyand thus in some cases the phase behavior of polymerblends.(7 – 10) Isotopic polymer mixtures have proven to bevery useful in assessing the widely used Flory–Huggins
theory,(11) because they lack the structural asymmetriesand specific interactions encountered in blends ofchemically distinct species. The miscibility of isotopicmixtures extends beyond the range predicted by theFlory–Huggins theory(9); that is, mixtures of isotopesare not ‘simple’ mixtures.
The last two factors in Equation (3) determine the angledependence of the scattering. P(Q) is the single-particleform factor, due to interference of waves scatteredby different parts of the same body. (Note in somepublications, F(Q) is used rather than P(Q).) Formfactors are shown in Figure 2 for some common particleshapes. S(Q) is the structure factor, determined byinterference among waves scattered from different lociwithin the sample. S(Q) tends to unity for small particleconcentrations and large scattering angles.
For low concentrations of scatterers, the scatteringintensity is governed primarily by the form factor;however, for higher concentrations, interparticle corre-lations become significant. At least for monodisperseparticles, I (Q) can be expressed as the product of P(Q),describing the particle shape, and S(q), describing theparticle positions. An example is given in Figure 3.(12)
The spatial range with neutron scattering varies froma few angstroms to as much as ∼105 A for USANS;this broad span makes the technique ideal for studyingpolymers and blends. Not only can measurements becarried out on blends in the homogeneous one-phaseregion, but also the kinetics of the phase separationprocess can be investigated, including information aboutthe resulting morphology and domain sizes.
00.0
0.2
0.4
0.6
0.8
1.0
1 2 3 4
RgQ
P(Q
)
Guinier
Chains
Rods
Spheres
Tubes
Figure 2 Common form factors for spheres, rods, tubes (i.e.hollow rods), Gaussian chains, and the Guinier approximation(Equation 6) valid at low angles versus the product of the radiusof gyration and momentum transfer.
To measure the dynamics of a material, inelasticneutron scattering is employed.(13,14) The techniquesinclude quasielastic neutron scattering (QENS) andneutron spin echo (NSE) experiments. In QENS, time-of-flight or backscattering spectrometers are used. As itsname implies, in a time-of-flight experiment, neutronsscattered from the sample are collected as a function oftime by wide-angle detectors. Backscattering experimentsuse a monochromator and analyzer crystals to resolve the
Figure 3 Illustration of particles in direct and reciprocal space.The SANS particle scattering profile is the product of theform factor times the structure factor. The asterisk denotesconvolution (from the study by Papon et al.(12) with permission).
energy of the scattered neutrons. NSE experiments takeadvantage of the magnetic moment of a neutron, whichcauses its precession at the Lamor frequency when in amagnetic field. This precession functions as an internalclock, providing precise determinations of the energydifference before and after scattering from the changes inneutron polarization with time.
3 ANALYSIS OF SMALL-ANGLE NEUTRONSCATTERING FROM POLYMERS
The distinctive feature of polymer molecules is theirrange of length scales,(15) and these can be probed indifferent regions of Q. However, this means that elasticscattering data must be analyzed in a number of waysto extract all the information. Many of the analyseswere originally developed for SAXS or light scattering,but can be applied to SANS with minor modifications.Inelastic neutron scattering applied to dynamics is less
common in polymer research, although it can be animportant supplement to more traditional methods, suchas dynamic mechanical and dielectric spectroscopies. Theranges of energy (10−8 –1 eV) and momentum transfer(0.1 to ∼50 A−1) for inelastic neutron scattering are verybroad, and thus its applicability as a dynamical probe isdiverse; for example, molecular vibrations (large �E andlarge Q), rotational tunneling (small �E and large Q),aggregate motions in biological systems (small �E andsmall Q), and glassy dynamics (large �E and small Q).
3.1 Single-Phase Systems
For a one-phase system, a power series expansion can beused for the form factor:
P(Q) = 1 − R2gQ
2
3+ R4
gQ4
12− · · · (4)
in which Rg is the radius of gyration. Through the first twoterms, this equation is the same as the series expansionof the exponential function; thus, the intensity at smallscattering angles can be written as
I (Q) ∝ exp(
−13R2
gQ2)
(5)
This is the Guinier approximation,(16,17) which providesa model-independent means to determine the radiusof gyration of discs, rods, rings, linear and branchedmacromolecules, etc. For spherical particles, Guinierplots (logarithm of intensity versus Q2) are linearup to QRg < 1; for elongated particles, the range ofvalidity extends to higher angles (typically QRg < 1.3 forproteins), depending on how the particular shape affectsthe Q4 term in Equation (4).
An alternative is the Zimm approximation,(18) whichincorporates the concentration dependence by makingmeasurements for a number of concentrations and doublyextrapolating to zero concentration and zero Q:
limQ,ϕ→0
ϕ
I (Q)= 2A2ϕ + V −1
m
(1 + R2
gQ2
3
)(6)
In this equation Vm is the molar volume and A2 thesecond virial coefficient characterizing intermolecularinteractions. Zimm plots are usually linear over a largerrange of Q than the Guinier plot. The intercept of aZimm plot provides the (absolute) molecular weight,which can serve as a check on the validity of the scatteringdata. For measurements that do not extend to sufficientlysmall Q, an alternative ‘generalized Zimm plot’ has beenapplied(19,20):
Figure 4 (a) Guinier plots and (b) Kratky plots for SANS data on two arborescent polystyrenes. Different solvents were used toselectively contrast match the inner or outer portions of the molecule, to obtain Rg for the shell and core, respectively. The twomethods of analyzing SANS data gave equivalent results (from the study by Yun(23) with permission).
I (Q) = ϕ(1 − ϕ)
[VmP(Q)]−1 + 2A2ϕ(7)
Scattering data from several concentrations are simulta-neously fit to Equation (7), with a model function usedfor the form factor P(Q).
For Gaussian chains (flexible, linear macromoleculesexhibiting a Gaussian distribution of segment densities),the scattering pattern depends only on the radius ofgyration and is described by the Debye function(21)
P (Q) = 2R4
gQ4
[exp(−R2gQ
2) + R2gQ
2 − 1] (8)
By measuring two samples having different levelsof isotopic substitution, the single-chain scatteringcontribution can be obtained without going to the limitof zero concentration. The measured coherent scatteringfor each sample is normalized by the respective scatteringlength, to yield the structure factor. For small Q (QRg <<
1), Equation (8) is equivalent to Equation (6); forintermediate Q (QRg > 1), Equation (8) gives
P(Q) ≈(
1 + R2gQ
2
2
)−1
(9)
Thus, a plot of I (Q)Q2 versus Q [Kratky plot(22)] shouldasymptotically approach a plateau value, which yields themolecular size. This analysis is often applied to determinethe conformation of proteins (e.g. their folding). Ifthe polymer has a non-Gaussian distribution of chainlengths, such as for branched or star polymers, theKratky plot deviates from the plateau. Figure 4 compares
Guinier and Kratky plots for arborescent (graft dendritic)polystyrene.(23)
3.2 Multicomponent Materials
According to the random phase approximation (RPA),(11)
the coherent SANS intensity due to concentrationfluctuations (assuming incompressibility and thereforeno density fluctuations) is
I (Q)−1 = �ρ−2[
1ϕ1ν1N1P1(Q)
+ 1ϕ2ν2N2P2(Q)
− 2χ√ν1ν2
](10)
where ν is the monomer volume, Ni the degreeof polymerization, and the contrast factor is permonomer volume. P(Q) in Equation (10) is the single-chain scattering function (Equation 8); thus, the RPAexpresses the structure factor of a system of interactingchains in terms of those of a noninteracting (‘ideal’)system. Equation (10) enables determination of theinteraction parameter, χ, which is a measure of themixing enthalpy plus any noncombinatorial entropycontributions to the free energy (the latter arising, forexample, from equation of state effects due to sizedifferences between components). Reflecting limitationsof the RPA description, experimentally χ is not strictlyconstant, but can depend on temperature, molecularweight, composition, and even on Q.(24 – 26) An exampleof the application of the RPA equation to a polymerblend is shown in Figure 5.(27) The data are plotted in the
Figure 5 SANS intensity for isotopic mixture plotted in theOrnstein–Zernike form (solid line is linear fit), along withthe fit to Equation (10) (dashed line); the difference betweenthe analyses is only apparent at small Q. Inset shows theinteraction parameter determined from measurements at fourtemperatures, yielding χ = 8.3 × 10−3 − 3.6/T (from the studyby Tomlin and Roland(27)).
Ornstein–Zernike form(11):
I (Q)−1 = �ρ−2[a0 + a1Q2] (11)
valid at small Q. The intercept of Equation (11) yields theinteraction parameter directly.
For a phase-separated material having a randomdistribution of phases, the length scale of the compositionvariations, ξ, can be determined from the Debye–Buechefunction(28):
I (Q) = 8πϕ(1 − ϕ)�ρ2 ξ3
(1 + ξ2Q2)(12)
A plot of I (Q)−1/2 versus Q2 yields the correlationlength in the limit of small Q. Equation (12) can be appliedto high concentrations, where the domains are mutuallycorrelated.(29,30) This differs from the Guinier analysis(Equation 5) for isolated domains. The high resolutionof SANS, especially USANS, makes the technique moreuseful than SAXS for studying the large domains oftenencountered with polymeric materials.(31 – 35)
In the case of sharp boundaries, the coherent scatteringat higher angles has a Q−4 dependence in accord withPorod’s law(36):
I (Q) = 2π�ρ2SQ−4 (13)
where S is the surface area to volume ratio. Rough ordiffuse interfaces reduce the exponent in Equation (13)
10−510−1
100
101
102
103
104
105
106
107
108
109
10−4 10−3 10−2 10−1 10−0 101
Q (nm−1)
δΣ/δ
Ω (
Q)
(cm
−1)
Q−4
Q−4
SANSUSANS
Figure 6 SANS and USANS of a solution of poloxamertriblock copolymer (an amphiphilic polymer commerciallyknown as Pluronic®). The various Q-dependences at differentscattering angles are denoted by the solid lines, with two Porodregions (from the study by Sharp et al.(37) with permission).
to less than 4. Figure 6 shows SANS and USANS data, inwhich two Porod regions are observed correspondingto scattering from polymer micelles at high Q andagglomerated micelles at smaller Q.(37)
By combining WANS, SANS, and USANS, anenormous span of scattering vectors can be probed.Figure 7 is an example of neutron scattering datacombining the three experiments, with various analysesapplied to the different Q ranges.(38)
10−410−2
10−1
100
101
102
103
104
105
10−3 10−2 10−1 100 101
Q (Å−1)
Qm
QmOrnstein−Zernike function
I(Q) =I(0)
1 + ξ2Q2
−4
Porod′ lawBragg peak
(101) (101)
U-SANS
SANS
WANS
×LS
51 min137 min174 min
I(Q
) (a
.u.)
Figure 7 Swollen polyvinyl alcohol gels. Along with thephase-separated domains, the cross-links give rise to crystallineBragg maxima at Q = Qm (Kanaya et al.(38) with permission).
3.3 Small-angle Neutron Scattering and MolecularMotions
Sections 3.1 and 3.2 pertain to elastic experiments, inwhich there is no energy discrimination of the scattering.However, experiments can also measure the number ofneutrons scattered that exchange energy with the sampleby an amount dE; this yields the double differentialscattering cross-section:
∂2�
∂�∂E= σinc
4πSinc(Q,ω) + σcoh
4πScoh(Q,ω) (14)
In Equation 14, σinc and σcoh are the respective incoherentand coherent cross-sections, and S(Q,ω) are the dynamicstructure factors.
The Fourier transform of Sinc(Q,ω) gives the inter-mediate scattering function, Sinc(Q, t), from which theself-part of the van Hove correlation function(39) can beobtained. (The van Hove function gives the probability offinding a particle i at position r at time t when there is aparticle j at the origin at time 0; the self-part means i andj are the same particle.) The frequency is given by thePlanck relation, ω = E/(2πh). Under the usual conditionof small energy changes, Q is close to the correspondingvalue for elastic scattering, = 4π/λ sin θ/2.
Figure 8 shows quasielastic intensities, I (Q,ω),measured in the frequency domain. The motion of thescattering nuclei causes broadening from the profile ofthe elastic background.(40) The Fourier transform of thesedata yield the I (Q, t) shown in Figure 9.(40) The broaderpeaks, reflecting faster motions, result in a more rapiddecay of the correlation of the nuclei positions over timerelative to their positions at an initial time 0.
Although experimental methods such as dielectric spec-troscopy, dynamic light scattering, mechanical measure-ments, and nuclear magnetic resonance (NMR) are moreprevalent in studies of molecular and polymer dynamics,neutron scattering methods offer certain advantages. Inaddition to the ability to enhance and tailor the contrastof the sample constituents, the QENS and NSE methodsprovide spatial information, because the scattering canbe measured as a function of wave vector. The utilityof this property is illustrated in a study of methacrylatepolymers, in which motion of the pendant group wasresolved from the main-chain dynamics by measuring thescattering at different values of Q (Figures 10–12).(41)
QENS and NSE measurements interrogate thedynamics at very short times (<10−8 s). For slowprocesses, the static scattering can be measured as afunction of time. However, changes occurring over inter-mediate timescales are more difficult to monitor withSANS. One approach is to resolve the output of the 2-Dpositional detector over time, which allows dynamic data
−20
−1 0 1
−10 0 10 20Energy (μeV)
Energy (meV)
0.01
0.1
1
0.01
0.1
1PEO600-0% Na
PEO600-49% NaResolution
PEO600-0% Na
PEO600-49% NaResolution
S(Q
,ω)
/ S(Q
,ω) m
axS
(Q,ω
) / S
(Q,ω
) max
(b)
(a)
Figure 8 Frequency domain QENS data from two spectrom-eters: (a) 0.4–50 ps temporal range; (B) 250–2500 ps range,both at Q = 1.04 A−1. The sample was a polyethylene oxideconducting polymer with 0% (filled symbols) and 4% (hollowsymbols) ion content. The presence of ions reduces the chainmobility (from the study by Sinha and Maranas(40) withpermission).
to be obtained for processes having time constants on theorder of 0.1 s.(42)
4 ELASTIC SMALL-ANGLE NEUTRONSCATTERING APPLIED TO POLYMERICMATERIALS
Analysis of the angle dependence of neutron scatteringprovides structural and morphological information, andthere is an enormous literature describing the applica-tion of SANS to polymeric materials. Selected aspectsare described later to illustrate the broad utility of thetechnique.
Figure 9 Inverse Fourier transformed data from Figure 8. The faster segment motion in PEO lacking added ions (filled symbols)causes a more rapid decay of the self-intermediate scattering function (from the study by Sinha and Maranas(40) with permission).
0 0.5 1 1.5 2
Q(Å−1)
I
II
PBMA-d14
∂σ/∂
Ω(a
rb. u
nits
)
375K350K330K310K275K200K150K100K
Figure 10 Differential scattering cross-sections measured fordeuterated polybutylmethacrylate. The two peaks at Q ≈ 0.5and 1.3 A−1 correspond to correlations between units in the mainchain and between atoms located in side groups, respectively(from the study by Arbe et al.(41) with permission).
4.1 Determination of Molecular Parameters
The scattering at large Q reflects the short-range,intramolecular correlations of the polymer segments.As shown from scattering measured for polybutadieneat three temperatures spanning the glass transition(Figure 13(43)), the static structure factor for Q > 5 A−1
remains constant. However, glass formation affects theposition and intensity of peaks at small values of Q
(<2.5 A−1), due to the significant change in molar
010−2 10−1 100 101
t(ns)
T = 480 K
PHMA
PBMA
PEMA
PMMA
0.2
0.4
0.6
0.8
1
S(Q
,t)/
S(Q
,0)
Figure 11 Dynamic structure factor for methacrylate poly-mers with different side groups, measured at the maximum atlower angle (peak I at 0.5 A−1 in Figure 10). The dynamicscorrespond to structural relaxation of the main chain (from thestudy by Arbe et al.(41) with permission).
As seen from the equations in Sections 3.1 and 3.2,detailed analysis of the Q dependence of SANS canyield information on the polymer chain conformationand also the interaction parameter of mixtures. Usingthe Guinier (Equation 5), Debye (Equation 8), or RPA(Equation 10) equations, the radius of gyration of thepolymer can be calculated (as seen in Figure 2, thescattering function for a Gaussian coil depends only on
Figure 12 Comparison of the dynamic structure factors atthe higher angle maximum (peak II at 1.3 A−1 in Figure 10), forthree methacrylate polymers (filled symbols) and a polyethylene(open circles). The NSE behavior reflects structural relaxationof the side groups (from the study by Arbe et al.(41) withpermission).
the radius of gyration). The single-chain structure factorfor polyisoprene extracted from SANS measurements ontwo samples differing in deuteration level is shown inFigure 14.(44) The fit to Equation (8) yields Rg = 34 A.Applying Equation (5) to the same data in the limit ofzero Q yields Rg = 31 A.
00.0
0.5
1.0
1.5
2.0
5 10 15 20
Q (Å−1)
Q (Å−1)
S(Q
) (a
rb. u
nits
)
Sf (
Q)
1,4-Polybutadiene
Tg = 176 K
0
1
2
2 4
257 K
168 K
25 K
Figure 13 Static structure factor from SANS for polybuta-diene measured in the liquid state (dashed line), just below Tg(dotted line), and deeply in the glassy state (solid line). Thesmall Q scattering is shown on an expanded scale in the inset(data from the study by Frick et al.(43)).
The branch structure of polymers is difficult to quan-tify by conventional methods; however, the results fromscattering experiments to characterize the chain size canbe combined with NMR or other spectroscopic deter-minations of (number average) branch concentration toobtain detailed information, especially when measure-ments are made on various molecular weight fractions.Figure 15(45) shows the persistence length, lp, from SANS
0.0001
2
3
4
5
6
7
0.002 0.004 0.006 0.008 0.010
q2 (Å−2)
S (
q)
(au)
Rg = 31 Å
Rg = 34 Å
Polyisoprene
M = 10 kg mol−1
Figure 14 Single-chain structure factor for a blend of deuterated and normal polyisoprene, plotted in the Guinier form. The solidline is the fit to the Debye equation, and the dashed line the Guinier fit to the low Q data (from the study by Akcasu et al.(44)).
Figure 15 Persistence length of linear low-density polyethylene chains versus the number of butyl branches. The line is the fit ofEquation (15) (data from the study by Ramachandran et al.(46)).
plotted as a function of the number of branches per thou-sand carbon atoms (determined from NMR). The dataare fit to an empirical expression relating lp to the numberbranches(46):
lp = l∞p − k1 exp(−n/k2) (15)
in which l∞p is the persistence length for completebranching, and k1 and k2 are constants, the former specificto the polymer and the latter depending on the type ofbranches. For polyethylene, l∞p = 9.1 ± 0.4 A.(46)
The formation of polymethymethacrylate (PMMA)star polymers by linking linear chains was followedby in situ SANS measurements carried out as afunction of reaction time.(47) As shown in Figure 16,the time evolution of characteristic structural param-eters can be extracted from analysis of the scatteringprofiles.
SANS is commonly used to study phase behavior andthe morphology of block copolymers and blends.(48,49)
In principle, both density and composition fluctuationscontribute to the small-angle scattering. By selectiveisotopic labeling of the components, the two effectscan be characterized independently. This is illustratedin Figure 17 for a blend of polyethylene oxide (PEO)and PMMA.(50) As the PEO is deuterated, the scatteringis mainly from composition fluctuations, which increasewith temperature as the LCST (lower critical solutiontemperature) is approached. The scattering due to densityfluctuations, determined by measurement on a completelyprotonated sample, is indicated in the figure.
00.06
0.10
0.14
0.18
0.220.6
0.7
0.8
0.9
1.0
1.168
10
12
14
16
18
20
10 20 30 40 50Polymerization time, t (h)
ηR
core
(nm
)N
SA
NS
Solution ASolution B
Solution ASolution B
Solution ASolution B
(a)
(b)
(c)
Figure 16 (a) Number of arms per molecule, (b) radius of thecore, and (c) volume fraction of the equivalent hard spheres forPMMA star polymers, as determined from time-resolved SANSprofiles of the reaction solution (from the study by Terashimaet al.(47) with permission).
S2: d-PEO0.25/h-PMMA0.75 140 °C101 °C80 °C71 °C62 °C54 °C24 °C
Figure 17 SANS from a blend of deuterated polyethyleneoxide and protonated polymethylmethacrylate. The upturn atlow Q is due to density fluctuations, whose contribution wasdetermined from SANS on blends without isotopic contrast (andindicated by the scattering data below Q ≈ 10−2 A−1). Note theOrnstein–Zernicke Q−2 dependence at high Q (Equation 11)(from the study by Schwahn et al.(50) with permission).
4.2 Phase-separated Materials
The phase separation process can be followed usingSANS, with nucleation and growth versus spin-odal decomposition mechanisms easily distinguished(Figure 18(51)). The phase separation process in polymersis slow enough that time-resolved scattering experimentscan be performed. For spinodal decomposition, the lengthscale, lm, of the phase-separated morphology produces amaximum in the scattering curve at a scattering anglegiven by the Bragg formula, Qm = 2π/lm. If the phaseseparation proceeds by nucleation and growth of domains,the morphology is too irregular to yield such maxima,although there is an increase in the scattering intensity(‘cloud point’). This can be assessed from the scatteringinvariant:
I =∫ ∞
0Q2 d�
d�(Q)dQ (16)
In calculation of scattering invariants, the integrationextends only to angles for which there is measurable inten-sity in the background-corrected scattering; however, fora two-phase system, this scattering at high Q usuallymakes a negligible contribution to I .
Block copolymers, in which the chain moleculescomprise multiple sequences of repeat units, are animportant class of materials with widespread commercialapplications. When the blocks are thermodynamicallyimmiscible, phase segregation occurs, giving rise to
4 5 6 7 8 9 2 30.1
q (nm−1)
101
102
103
104
105
106
107
108
I (cm
−1)
170 °C
130 °C
120 °C
110 °C
30 °C
Figure 18 SANS intensity of a blend of saturated 1,2-polybu-tadiene and polyisobutylene with added diblock copolymer as acompatibilizer. At temperatures below the critical temperature,120 °C, the lamellar structure gives rise to peaks at Qm and2Qm. At higher temperatures, the morphology becomes homo-geneous and the peaks are absent (from the study by Nedomaet al.(51) with permission).
morphologies ranging from spherical, cylindrical, gyroid,or lamellar, as governed by the block lengths andmatrix.(52 – 55) SANS has been applied extensively to inves-tigate the morphology of block copolymers. In Figure 19are SANS data for styrene–isoprene diblock copolymerin which the styrene was fully deuterated. Peaks corre-sponding to successive orders in the interference patternare evident.(56)
Figure 20 compares the SANS and SAXS patternsfrom a triblock polymer.(57) The diffraction intensitiesare calculated from the Fourier coefficients of thesegment densities, each term corresponding to a particularreflection.(58)
The distribution of deuterated water in epoxy hydrogelswas studied using SANS.(59) Two regimes for themorphology were observed: below the percolationthreshold of water, isolated water-rich domains aredispersed in a continuous water-poor epoxy matrix; athigher water content, these domains become continuous.With continuity of the dispersed phase, there is adiminution of the scattering maxima (Figure 21).
Figure 19 SANS from an isoprene–styrene diblock copol-ymer. The arrows denote the scattering vectors of the variousBragg diffraction peaks. The most intense at Q ∼ 0.008corresponds to a domain spacing of about 800 A (Richards(56)
with permission).
0.01 0.02 0.03 0.04 0.05
q (Å−1)
q (Å−1)
SAXS
SANS
Inte
nsity
(a.
u.)
Inte
nsity
(a.
u.)
0.01 0.02 0.03 0.04 0.05
(a)
(b)
Figure 20 Small-angle X-ray scattering and SANS diffractionpatterns from an isoprene–styrene–dimethylsiloxane triblockcopolymer. The histogram bars are the calculated intensitiesof the various peaks assuming a core-shell gyroid morphology.The first-order reflections are cancelled in the SANS spectra(Shefelbine et al.(57) with permission).
Polymer composites, in which filler particles are usedto reinforce or otherwise modify the properties, are verycommon materials. The resolution of SANS and USANSis advantageous for characterizing particles sizes,(60 – 63)
including rheo-SANS experiments that probe the effectof deformation on the particulate structure(12,64 – 66)
(Section 4.3). Chu and Liu(67) compared light, X-ray,and neutron scattering methods for determination ofthe size of particulate fillers in polymers. Figure 22shows SANS curves measured for polystyrene/silicananocomposites.(68) After normalizing the intensity bythe filler content, the curves superpose at high angles,with agglomeration of the silica giving rise to a peak atsmaller scattering angles, at a value of Qm that varies withthe 0.87 power of the filler volume fraction.
Polyhedral oligomeric silsesquioxane (POSS) is oneof the smallest filler particles, with a symmetric, three-dimensional nanostructure. In Figure 23(69) are SANSdata for POSS suspended in deuterated toluene.
A study of polyurea ceramic composites using WANSmeasurements was carried out (Figure 24).(70) The smallpore size of the ceramic required the wider anglemeasurements, which provide information similar toX-ray scattering, but with different contrast.
4.3 Effect of Strain and Rheo-small-angle NeutronScattering
The ability of SANS measurements to provide structuralinformation makes it useful for investigating the effectof deformation and flow on materials. The greaterpenetrating power of neutrons is a key advantage ofSANS over SAXS or light scattering for such experiments.Also, contrast matching can be employed to probe theorientation or positional ordering of specific componentsof the sample. For solids or materials with a stablemorphology, the scattering can be measured subsequentto the deformation, although in situ measurements madeon flowing samples (rheo-SANS) is an increasinglypopular technique.
Phase-separated polymeric materials, including gels(cross-linked rubbery polymers in solution), producecharacteristic scattering when deformed known asbutterfly patterns. These are isointensity contours on a2-D detector that have a double-winged shape, symmetricalong the stretch direction.(48,71 – 74) Shibayama(75) hasreviewed the various butterfly patterns characteristic ofdifferent types of gels. An example for a sheared blend isshown in Figure 25.(76)
Figure 26 shows the SANS and SAXS intensitiesparallel and perpendicular to the orientation directionfor drawn polyethylene (Mw = 2 × 106 g/mol) containinga few percent of deuterated PE (Mw = 2 × 105 g/mol).Using the SAXS data to subtract the contribution fromdensity fluctuations, the resulting scattering patternswere fit to a model to obtain a quantitative descriptionof the shish-kebab, extended chain semicrystallinemorphology.(77)
Figure 21 Coherent SANS intensities for swollen epoxy networks (heavy water concentrations by weight = 0%, 23%, 47%, 58%,67%, 72%, and 77% from bottom to top). Schematics are shown for the structure below and above the concentration for percolationof the water phase (from the study by Krakovsky et al.(59) with permission).
0.0010.001
0.01
0.1
1
10
100
1000
104
105
0.01 0.1 1
Q (Å−1)
Q (Å−1)
Sca
tterin
g in
tens
ity /
%v/
v S
iO2(c
m−1
)
00
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Iq3
6.65% v/v
10.64% v/v
15.43% v/v
19.75% v/v
29.4% v/v
Figure 22 Coherent SANS for PS with differ amounts ofsilica particles. At high Q, the intensity deceases as Q−4,reflecting the sharp interface between the polymer and thesilica (Equation 13). The inset shows the oscillations in IQ3 inthe region around Q = 0.04 A, due to internal structure withinthe silica aggregates (from the study by Jouault et al.(68) withpermission).
An example of following phase separation by makingSANS measurements as a function of time is shown inFigure 27 or a blend of PS and PMMA.(78) Deformationaffects the phase separation behavior; thus, the scattering
0.01
0.01
0.1
Q (Å−1)
Q2 (Å−1)
dΣ/d
Ω (
cm−1
)
In d
Σ/dΩ
(cm
−1)
0.1
0.100.080.060.040.02
−4.0
−3.5
−3.0
R R
R
R
RR
R
O
O
O
O
O
O
O
OO
O
O
OSi
Si
Si
Si
Si
SiSi
SiR
Figure 23 Form factor for POSS particles (structure shown) indilute toluene solution. Guinier plot in the inset yields Rg = 6 A(from the study by Nusser et al.(69) with permission).
patterns differ after imposition of small and large shearstrains. The latter accentuates phase separation, withconsequent increase in the scattering intensity and a shiftof the maximum to smaller angles.
The opposite of phase segregation, coalescence ofdispersed polymer phases, can also be quantified usingSANS. Phase-separated blends usually consist of discreteparticles in a continuous phase. For an immiscible blend,this globular morphology is never at equilibrium; rather,the size distribution of the dispersed phase is a steady-state balance between the breakup of the particles
Figure 24 WANS from polyurea polymers synthesized within the pores of a ceramic (the two curves correspond to different ratiosof isocyanate to polyamine). The ordering of the hard domains gives rise to crystalline-like reflections, denoted in the figure (fromthe study by Zaj1c et al.(70) with permission).
ω = −84° ω = −81°
qx
qyqz
q-Region (I) q-Region (I)
q-Region (II) q-Region (II)
Direct beam
(a) (c)
(b) (d)
L1 L2 L1 L2
L1 L2 L1 L2
q(A−1)0.003 0.035
q(A−1)0.035 0.15
Weak WeakStrong Strong
′
′′
Figure 25 SANS profiles of a blend of polystyrene andpolyvinylmethylether for shear rotation of the indicatedmagnitudes. The upper curves are the low Q range at whicha double-lobe pattern is observed. The lower curves at higherscattering angle exhibit an elliptical shape. The directions ofshear, velocity gradient, and vorticity are x, y, and z, respectively(from the study by Koizumi and Suzuki(76) with permission).
and their coalescence. This flow-induced coalescencewas followed for polybutadiene particles dispersed ina polychloroprene matrix from the change in the
0.00110−1
100
101
102
103
104
105
106
107
0.01
Q (Å−1)
0.1 1
SAXS
SANS
PE-d4/PE-h4 (97.2/2.8)
Inte
nsity
(a.
u.) SAXS_parallel
SAXS_normal
SANS_normalSANS_parallel
Figure 26 Scattering from high molecular weight polyethy-lene containing 2.8% deuterated PE measured along andtransverse to the elongation direction (Kanaya et al.(77) withpermission).
SANS invariant (Equation 16) with time of shearing(Figure 28).(79) Initially, the particles were either whollydeuterated or protonated polybutadiene; coalescencescauses particles to have a mixed isotopic composition,which reduces the scattering intensity.
The rheo-SANS method involves combining flow of thesample with simultaneous scattering measurements.(80)
These experiments employ various flow types andgeometries, including shear flow in concentric cylinders(Couette cell), sliding plates, or extensional flow using,for example, capillary tubes or flow from opposingjets. The anisotropy of the flow field defines differentscattering planes, as illustrated in Figure 29 for a Couette
Figure 27 SANS and USANS measured along the shear(x) and vorticity (z) directions following shear-induced phaseseparation from a blend of PS and PVME (S indicates the valueof the shear strain). The scattering at high angle follows Porod’slaw (from the study by Koizumi and Inoue(78) with permission).
0
0
20
40
60
80
100
10 20 30 40 50 60Mill passes
Coa
lesc
ed p
artic
les
(%)
CollisionFilm
draining
Film rupture Coalescence
5% PBDin polychloropene
T = 80 °CShear rate ≈ 80 s−1
Figure 28 Amount of polybutadiene dispersed in polychloro-prene that has coalesced with other dispersed particles versus theextent of mixing (i.e. number of passes through a two-roll mill)at the indicated conditions. The mean particle radius was 40 nm.Coalescence was reflected in the reduction in scattering inten-sity. The inset shows an idealized depiction of shear-inducedcoalescence of dispersed particles (from the study by Rolandand Bohm(79)).
geometry (sample sheared in the gap between twocylinders).(81)
33
1
3
2
1
2
2
1
2-D scattering planes
Neutroncross-section
(a) (b) (c) (d)
Figure 29 Couette flow geometries showing the differentscattering planes. 1 is the flow direction, 2 the direction ofthe velocity gradient, and 3 the vorticity direction (Eberle andPorcar(81) with permission).
Flow-induced anisotropy in a poloxamer (poly(ethylene glycol)–poly(propylene glycol)–poly(ethyleneglycol) triblock copolymer, an amphiphilic material)sheared in a cone and plate device is seen in the transitionof Debye–Scherer rings in the scattering pattern fromthe quiescent material to sharp reflections (Figure 30).(82)
Figure 31 compares for a similar material in deuteratedwater the scattering in the radial direction (the usualmeasurement geometry) with SANS in the tangentialdirection.(83) The tangential scattering provides informa-tion about the stacking of the 2-D layers.
SANS of concentrated aqueous solutions of polox-amers exhibit peaks corresponding to cubically packedspherical micelles (Figure 32).(84) The domainmorphology and hence scattering patterns show hysteresisduring traversal of the order–disorder transition.
Rheo-SANS was used to determine the radius ofgyration of entangled polystyrene in the nonlinear flowregime (Figure 33).(85) The sample was subjected torecirculating flow through a slit, with the scatteringmeasured both parallel and perpendicular to the flow.
SANS is an attractive technique for characterizingthe complex, hierarchical structure of biopolymers,particularly in their natural environment. The numberof such studies, including rheo-SANS experiments, hasproliferated in recent years.(86 – 92) Plotted in Figure 34are various properties of fibrin, a highly branched, cross-linked mesh integral to blood clotting, determined usingrheo-SANS.(93)
4.4 Neutron Reflection
Less common than neutron scattering, neutron reflection(NR) is used to probe the structure normal to aninterface.(94 – 97) The neutron reflectivity, R(Q), definedas the ratio of the reflected and incident elastic scatteringintensities, is given to a good approximation by(94):
Figure 30 SANS from a poloxamer in transmission mode: (top) no flow (bottom) shearing at 100 s−1. The annotations identifythe Debye–Scherer reflections from the face-centered-cubic morphology (from the study by Wolff and Zabel(82) with permission).
where F represents the Fourier transform of the scatteringlength density profile and z is the dimension perpendicularto the surface. In practice, analyses of NR experimentsrely on fitting models to the data, which means thesolutions are not always unique. R(Q) is measured as afunction of Q, typically over a range of ca. 0.01–0.2 A−1,obtained by changing either the neutron wavelength orthe grazing angle. The Q−4 dependence (Fresnel’s law)
in Equation (17) assumes a perfectly smooth interface;surface roughness causes R(Q) to decrease more stronglywith angle. This can limit the attainable Q, but infavorable cases provides information about the surfacetopography.(98)
The reflectivity yields information on layer composi-tion, thicknesses, and roughness as a function of depthperpendicular to the surface. NR is specular (neutrons
Figure 31 SANS radial (left) and tangential (right) patterns for a poloxamer solution sheared in a Couette apparatus at shearrates (a through i) equal to 0, 0.01, 1, 5, 10, 15, 30, 50, and 600 s−1 (from the study by Jiang et al.(83) with permission).
from a single incoming direction are reflected in asingle outgoing direction), and the reflected intensitydepends only on the scattering length densities; thus,the method cannot distinguish material variations fromphysical roughness. Diffuse (off-specular) NR can be
used to characterize the in-plane composition and lateralvariations of structure.(99)
NR finds application in biological materials, wherethe ability to contrast match is very important, giventhe complexity of the structures.(100) For example, the
0100
1000
60 120 180 240 300 360
Azimuthal position (°)
Inte
nsity
(cm
−1)
Q = 0.039 A−1
Primary ring, 10 °C to 45 °C, oscillation 5Hz 100%
Primary ring, 10 °C to 45 °C, shear 1s−1
45 °C 45 °C
e
v
e
v 5Hz, 100% 1 s−1
Figure 32 Rheo-SANS profiles and azimuthal intensities measured radially for a poloxamer solution under oscillation (2-Dpattern on left and filled symbols) and during steady flow (2-D pattern on right and open symbols). The hexagonally packed cylinderphase is present during oscillation but is disrupted by steady shearing (LaFollette and Walker(84) with permission).
Figure 33 Radius of gyration of polystyrene as a function of flow rate, determined from fitting the Debye function (Equation 8)to the SANS data. The anisotropy begins to level off at higher flow rates (Graham et al.(85) with permission).
ratio of protons and deuterons can be adjusted toeliminate the contribution of water. This is illustratedin Figure 35(101); the contrast was varied to characterizethe layers, with the amount of heavy water changedto selectively emphasizing different components of themembrane.
NR has been applied to the study of the inter-facial configuration of macromolecules with unusual
2000
3000
4000
50000.0
0.1
0.2
0.3
0.4
20
30
40
75
80
85
90
Strain (%)
GIn
st (
Pa)
1 10 1002 4 6 8 2 4 8 26
P2
φ 0 (
°)D
(nm
)
(a)
(b)
(c)
(d)
Figure 34 Diameter (a), alignment angle (b), order parameter(c), and instantaneous modulus (d) of fibrin fibers as a functionof shear strain (from the study by Weigandt et al.(93) withpermission).
architectures, such as brush polymers.(102 – 104)
Figure 36(104) shows a Porod plots (Equation 13) of NRdata for a polymer brush grafted onto a gold substrate andimmersed in heavy water. From the measured reflectivity,the polymer concentration profile was determined as
ϕ(z) = ρ(z) − ρsub
ρpoly − ρsub(18)
where ρpoly and ρsub are the respective scattering lengthdensities of the polymer brush and the substrate and z
is the distance from the surface. The total amount ofpolymer at the interface was constant, but its spatialdistribution varied in accord with its compatibilitywith water. The obtained depth profiles are shown inFigure 37.(104) As the temperature is increased above theLCST, there is partial phase separation normal to thesurface, resulting in a denser inner phase and a diluteouter phase.
A similar approach can be used to measure theinterdiffusion at the interface of two polymers.(105 – 107)
Model calculations are carried out that yield ρpoly that fitthe experimental reflectivities. Results for interdiffusionof polystyrene with its deuterated analog are shown inFigure 38.(107)
Figure 39 shows the interfacial profile for a bilayerof 1,4-polybutadiene with various rubbers.(105) The peeladhesion measured for the different adherends increasesin proportion to the extent of interpenetration, the lattermeasured by NR.
Other applications of NR include investigating blockcopolymer morphologies,(108 – 110) the mobility of polymersegments in confined geometries,(111,112) and exam-ining the interface and adhesion of polymers onsubstrates.(113,114) Instruments to carry out in situ NR onsamples undergoing shear flow have been developed.(115)
Figure 35 Neutron reflectivity versus momentum transfer for hybrid bilayer membranes with three contrast levels: D2O (lowercurve), 50/50 D2O/H2O (middle curve), and H2O (upper curve). The inset is the scattering length density obtained from modelingthe reflectivity data (from the study by Smith et al.(101) with permission).
Figure 36 Porod plots of neutron reflectivity measured frompolymer brush grafted to a gold substrate as a function oftemperature. Solid lines are fits to the NR data. Curves arevertically displaced for clarity (from the study by Jia et al.(104)
with permission).
4.5 Polymer Dynamics
Figure 40 compares the coherent dynamic structure factorfor polyethylene oxide rings, linear chains, and their
Figure 37 Interfacial polymer profiles determined from thedata in Figure 36 for various temperatures. The LCST is between30 and 40 °C (from the study by Jia et al.(104) with permission).
mixture at Q = 0.05 A−1.(116) The faster decay for rings isconsistent with their lower melt viscosity.
As stated, an important advantage of neutron scatteringover more conventional relaxation spectroscopies is theability to measure the dynamics at different lengthscales. Figure 41 shows the change in the breadth ofthe incoherent QENS spectra with scattering vectorfor polyethylene.(117) Small Q measurements probe thedynamics within the ‘cage’ (first coordination shell) ofneighboring atoms, while the higher Q data are sensitiveto the chain dynamics.
The motion of polyethylene oxide chains adsorbed onclay was monitored using NSE.(118) The data (Figure 42)
Figure 38 (top) NR profiles for various contact times,vertically offset for clarity; (bottom) model scattering lengthprofiles for a polystyrene/D-polystyrene bilayer film (from thestudy by Kawaguchi et al.(107) with permission).
conform to Zimm scaling, according to which therelaxation time varies as(119)
τ ∝ Q−3η/T (19)
where η is the viscosity. The data measured at differentscattering vectors collapse to a single master curve whenplotted versus the produce of time and Q3.
5 SUMMARY
Neutron scattering is a powerful analytical tool forinvestigating polymers. Unique aspects such as thecapacity to measure thick samples with an absenceheating or radiation-induced damage, selective controlof the scattering contrast by isotopic substitution, andthe measurement of dynamic properties as a functionof length scale are among the many attributes that
PDMS
EPDMSBR160KSBR36%Sty
PIB
z (Å)
–200 –100 0 100 200
φ PB
120K
-D
0.0
0.2
0.4
0.6
0.8
1.0
Figure 39 Volume fraction of deuterated polybutadiene atthe interface with various polymers, as determined by neutronreflectivity. The strongly incompatible polydimethylsiloxaneand polyisobutylene adherends form sharp interfaces (15- to30-A breadth), whereas the styrene–butadiene copolymer oflower styrene content gives a broad interface (∼200 A) withthe polybutadiene (from the study by Schach et al.(105) withpermission).
Linear 0.05 Å–1
Ring
Ring/linearmixture
S(Q
, t)/
S(Q
, 0)
0.4
0.6
0.8
1.0
t (ns)0 5 10 15 20 25
Figure 40 NSE scattering curves for different topologiesof polyethylene oxide, showing the faster dynamics of ringscompared with linear chains having the same molecular weight.The solid lines are fits to the Rouse model (from Bras et al.(116)
with permission).
make neutron methods not just complementary to otheranalytical techniques, but in many cases the only meansto address material issues. The information that can beobtained from neutron scattering – the size, form, andorientation of molecular chains, the thermodynamics andphase structure of mixtures, interfaces and interfacialphenomena, the dynamics at both the local and globallength scales, details of the structure of complexnanocomposites and biological materials, etc. – hasonly been surveyed herein. It is a testament to the
Figure 41 Time-of-flight neutron scattering spectra forpolyethylene at various Q and T = 477 K. The dotted lines indi-cate the instrumental resolution. The peak breadth is plotted inthe inset versus Q2; a broader peak signifies faster motion (fromthe study by Arbe and Colmenero(117) with permission).
0.0
10–5 10–4 10–3 10–2 10–1 100 101
0.2
0.4
0.6
0.8
1.0
8% PEO
4% PEO
τ Q3 (ns Å–3)
[S(Q
, τ)/
S(Q
, 0)−
f (Q)]/
(1−f
(Q)) τ × 100
Figure 42 Intermediate scattering function versus the productof time and the cube of the scattering vector for twoconcentrations of PEO/clay solutions in heavy water. Thesuperposition of the curves for different Q is consistent withZimm scaling (Equation 19) (from the study by Frielinghauset al.(118) with permission).
utility of neutron scattering that even though theapplication of neutrons to polymers did not beginin earnest until the 1990s, through the past decade,about 325 peer-reviewed publications per year haveappeared on the topic. Whether this figure changesin the future will likely depend on the availability ofthe specialized, costly facilities required for neutronexperiments. Certainly there is no expectation ofabatement in the number of important problems inpolymer science that can be addressed using neutronmethods.
ACKNOWLEDGMENTS
This work was supported by the Office of Naval Research.
Polymers and Rubbers (Volume 8)Polymers and Rubbers: Introduction
Polymers and Rubbers (Volume 9)Infrared Spectroscopy in Analysis of Polymer Struc-ture–Property Relationships • Infrared Spectroscopy inAnalysis of Polymers and Rubbers • Near-infrared Spec-troscopy of Polymers and Rubbers • Nuclear MagneticResonance, Solid state in Analysis of Polymers andRubbers • X-ray Scattering in Analysis of Polymers
Surfaces (Volume 10)Ellipsometry in Analysis of Surfaces and Thin Films
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