Neutron Scattering

2.11

Neutron Scattering

D Richter, M Monkenbusch, and D Schwahn, Jülich Centre for Neutron Science (JCNS), Jülich, Germany © 2012 Elsevier B.V. All rights reserved.

2.11.1 2.11.1.1 2.11.1.2 2.11.1.3 2.11.2 2.11.2.1 2.11.2.1.1 2.11.2.1.2 2.11.2.1.3 2.11.2.1.4 2.11.2.2 2.11.3 2.11.3.1 2.11.3.1.1 2.11.3.1.2 2.11.3.1.3 2.11.3.1.4 2.11.3.2 2.11.3.2.1 2.11.3.2.2 2.11.3.3 2.11.3.3.1 2.11.3.4 2.11.4 2.11.4.1 2.11.4.2 2.11.4.3 2.11.4.3.1 2.11.4.3.2 2.11.4.4 2.11.4.5 2.11.5 Appendix References

Introduction Coherent and Incoherent Scattering Coherent Scattering and Coarse Graining Contrast Generation and Variation Methods Small-Angle Neutron Scattering Intensity at sample and detector Pinhole SANS Toroidal mirror focusing SANS Double crystal diffractometer Spin Echo Spectroscopy Representative SANS Results Polymer Chain Conformation Homopolymer melts Melt of deuterated and protonated polymers with chemically identical monomers of same segment length H/D polymer blend in solution Scattering from diblock copolymer melts Binary Homopolymer Blends: Random Phase Approximation Phase separation and miscibility Deviation from Flory–Huggins theory: Critical phenomena Phase Behavior of (A/B) Polymer Blend/(A-B) Diblock Copolymer Mixtures Isotropic Lifshitz system Quenched SANS Polymer Dynamics The Rouse Model Reptation Reptation Limiting Processes Contour length fluctuations Constraint release Soft Confinement Dynamically Asymmetric Blends Conclusions

2.11.1 Introduction Thermal and cold neutrons have de Broglie wavelengths from λ = 0.1 to 2 nm corresponding to velocities of υ = 4000 m s−1 down to 200 m s−1. The wavelength range cov­ ers that of X-ray and synchrotron radiation diffraction instruments. However, in contrast to electromagnetic radiation, the neutron velocity has the same order of magnitude as the atomic velocities in the sample and the kinetic energy of the neutrons – in the order of meV other than typical X-ray energy of several keV – compares with the characteristic energies of atomic or molecular motions. Therefore, even the slow relaxational motions in polymers are detectable by a velocity change of the neutron. The spatial character of the motion can be inferred from the angular distribution of the scattered neutrons.

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In general, scattering of thermal neutrons yields information on the sample by measurement and analysis of the double differential cross section:1,2 d2 σðθÞ kf 1 X ¼ 〈bi bj 〉Si;j ðQ; ωÞ dΩ dE ki N i;j

½1

that is, the intensity of scattered neutrons with energy Ef into a given direction θ. The energy transfer, that is, the difference of kinetic energy before and after the scattering, ΔE = Ef – Ei relates to ℏω = ΔE. The momentum transfer ħQ, respectively the wave vector Q, is given by Q = k i – k f, where k i and k f are the wave vectors of the incoming and outgoing (scattered) neutrons. They relate to the neutron wavelength ki,f = 2π/λi,f; the neutron momenta are p i,f = mnυi,f = ħk i,f. The energy transfer ΔE can be determined by measurement of the neutron velocities υi and υf. Note that for all problems discussed in this chapter |k i| ≈ |k f|

doi:10.1016/B978-0-444-53349-4.00030-3

331

332

Structure Characterization in Fourier Space | Neutron Scattering

and therefore Q ¼ ð4π=λÞsinðθ=2Þ ¼ 2jki j sinðθ=2Þ can be assumed. This regime is also called ‘quasielastic’. Finally, bi denotes the scattering length of atom nucleus i and 〈…〉 is the ensemble average. The unique features of neutrons that render them a power­ ful tool for the investigation of ‘polymers’ are 1. the isotope and spin dependence of bi, 2. typical wavelengths of cold and thermal neutrons that match molecular and atomic distances, and 3. even slow motions of molecules cause neutron velocity changes that are large enough to be detectable, in particular neutron spin echo (NSE) spectroscopy is able to resolve changes Δυ of the order of 10−5υi. To proceed further, we introduce the intermediate scattering function S(Q, t) as the Fourier transform of S(Q, ω). S(Q, t) directly depends on the (time-dependent) atomic positions: �X � j i Sij ðQ; tÞ ≃ ei Q⋅½R n ðtÞ−R m ð0Þ ½2 n;m i

Note that, in general, the position of an atom n of type i, R n ðtÞ, is a quantum mechanical operator rather than a simple, time-dependent coordinate. For polymer investigations in the (Q, ω) range discussed here, ℏω  kBT, T ≈ 250–500 K, and Ei  Ebond, conditions for which R in ðtÞ may safely treated as classical coordinate and S(Q, ω) ≈ S(Q, –ω).

2.11.1.1

Coherent and Incoherent Scattering

Considering the ensemble average of eqn [1], we have to notice that chemically equivalent atoms may have a number of differ­ ent scattering lengths bi that are randomly distributed over the ensemble of all atoms of the same kind in the sample. Most important in the present context is the variation due to the spin-dependent component of the proton scattering length. Whereas the average value 〈bi〉 leads to coherent scattering, the fluctuating part 4πð〈b2i 〉 − 〈 bi 〉2 Þ leads to incoherent scattering introducing an additional contribution from the atom selfcorrelation: � 2 � X D iQ⋅ R i ðtÞ−R i ð0Þ E ð n Þ ≃ exp − Q �R 2 ðtÞ� n Sself ðQ; tÞ ¼ e ½3 i i 6 n where the right-hand side is a result of the Gaussian approxima­ tion that assumes Gaussian distribution functions for the atomic displacements. Note that the spin state of the scattered neutrons (spin-flip scattering) changes with a probability of 2/3.3

2.11.1.2

Coherent Scattering and Coarse Graining

Many polymer problems – including those discussed in this chapter – address the structure and dynamics in a mesoscopic regime. Here a description in terms of individual atom coordi­ nates R is not adequate and a coarse-grained description in terms of scattering length density Δρ(r , t) is used. To do so, a molecular unit of type j (e.g., a polymer segment, a monomer, or a whole smaller molecule) is selected and the sum of the scattering lengths of the contained atoms is related to the P effective volume of this unit, ρj = i∈j bi/Vj. Then the scattering only depends on the scattering length density difference, the

contrast Δρ(r , t) = ρpolymer – ρmatrix. To yield a valid description of the scattering, the extension of the chosen molecular unit L should be smaller than L < 1/Qmax. Then the related scattering function is ðD E 0 SðQ; tÞ ¼ Δρðr ; tÞ ⋅ Δρðr 0 ; 0Þ eiQ⋅ðr −r Þ d3 r ½4 The corresponding small-angle neutron scattering (SANS) intensity is proportional to S(Q, t = 0).4

2.11.1.3

Contrast Generation and Variation

The above description implies that contrast variation and matching can be employed to enhance or suppress the con­ tribution of a signal from selected subunits of a system. This is done by selective deuteration such that a contrast with respect to the generally hydrogen containing compounds of the sam­ ple is created. For solutions, normally deuterated solvents are used. NSE, in general, requires a deuteration of the matrix in order to optimize the intensity to background ratio. On the other hand, because of the very high incoherent cross section of hydrogen in quasielastic experiments aiming on the self-motion of the atoms, deliberate hydrogenation may be used in order to highlight certain molecules or molecular parts. Prominent examples of successful application of contrast variation are the investigations of the single-chain structure5 and dynamics of polymers in melts.6 Further details are obtained by investigating d-polymers that contain only a h-labeled section, that is, at the ends,7 at branching points or at its center in a fully deuterated matrix.8

2.11.2 Methods Two experimental methods, namely SANS and NSE spectro­ scopy, are presented here in some detail. Whereas NSE spectroscopy measures dynamic processes such as diffusion of mesoscopic large objects, SANS determines large structures of similar length scale and their kinetics during relaxation to equilibrium. Both techniques are complementary; SANS yields a scattering pattern that corresponds to the ensemble coverage of the conformation at one instant, whereas NSE spectroscopy allows the tempered change at the structure to be followed.

2.11.2.1

Small-Angle Neutron Scattering

The methods of elastic scattering with neutrons deliver structural information from arrangements of atoms and magnetic moments in condensed material. Those arrangements can be precipitated phases in mixtures of metals, low-molecular-weight liquids, or polymer melts. In scattering experiments, the inten­ sity of neutrons is measured as a function of momentum transfer Q from which information about size, number density, and correlation between the objects is derived. The momentum transfer is inversely proportional to the length scale of the inves­ tigated objects; at Q in the range of 10 nm−1, the method is sensitive to interatomic distances and in the range of 1–10−3 nm−1 to mesoscopic large objects of sizes between 1 and 103 nm. SANS instruments are optimized for the latter range of Q. We will discuss here three SANS instruments working

Structure Characterization in Fourier Space | Neutron Scattering

at stationary nuclear research reactors. In the future, spallation sources will become the more important sources as they show a larger neutron flux with a periodic time structure; those instru­ ments need different conditions for optimization.

Selector

Apertures

Sample

333

Detector

2θ

2.11.2.1.1

Intensity at sample and detector

In Figure 1, the traces of neutrons for elastic scattering are depicted in real and reciprocal space. The ‘intensity at the sample’ is determined according to ΔI0 ¼ L ⋅ F ⋅ ΔΩ by the ‘luminosity (cm−2 s−1 steradian−1)

of

the

source’

given

in

units

½6

for neutrons with wave vector k, the irradiated area of the sample F, and the divergency of the primary beam described by the solid angle ΔΩ. The luminosity L is determined by the total thermal flux of the neutrons Φ, the temperature of the moderator ðh=2πÞ2 k2T =2m ¼ kB T; and the resolution of wavelength distribution according to Δk/k determined by the monochromator. The scattered intensity in a detector element with solid angle ΔΩD and scattering angle θ or scattering vector Q is given as dΣ ðθ; QÞ ΔΩD dΩ

ΔID ðθ; QÞ ¼ ΔI0 DT

½7

with sample thickness D and diminution coefficient T of the primary intensity (transmission). The macroscopic scattering cross section (scattering cross section per unit volume) dΣ/dΩ is the experimental result and is given in absolute units (1 cm−1). The SANS method is a widely used tool in research. There are three different SANS techniques: the pinhole SANS, the double-crystal diffractometer (DCD), and the focusing SANS.

2.11.2.1.2

Pinhole SANS

The principle lay out of a pinhole SANS is depicted in Figure 2. After the fission process, thermalization, and a further modera­ tion in the ‘cold source’, the neutrons are guided through ‘neutron guides’ to the instrument. Monochromator and collima­ tor are filters for neutrons with predetermined wavelength and divergence. The collimator consists of two apertures of neutron absorbing material such as 10B and a monochromator, typically a ‘velocity selector’, is used that delivers a roughly monochromatic beam of wavelengths between 0.5 and 2 nm. The transmitted wavelength band is typically triangular shaped with a bandwidth δλ/〈λ〉 ≅ 10–20%. After passing both apertures, the neutrons hit the sample and some of the neutrons are scattered. The thickness of the sample should be adjusted in such a way that about 10% of Sample

ΔΩ, Δλ/λ

θ

F

θ k′

Figure 2 Principle design of pinhole SANS.

the neutrons are scattered in order to avoid remarkable effects from multiple scattering. The scattered neutrons are counted in a two-dimensional (2D) position-sensitive detector. The trans­ mitted neutrons remain in the primary beam and are absorbed in the beam stop in front of the detector. The resolution function of this experiment is given as4 "� � � �2 # � �2 � � � 2 � k2 dD 2 dE 1 2 2 δλ 2 1 δQ ¼ þ þ dS þ þθ LS LD 〈λ〉 12 LD LS ½8 where LS und LD represent the distances between the two apertures and between sample and detector, respectively, dD and dE the diameter of the two apertures, and δλ/〈λ〉 the bandwidth. For a given instrumental setting, neutrons can be detected in a limited angular interval; the setting is deter­ mined by the distance between sample and detector, usually between 1 and 20 m leading for the possible neutron wave­ lengths between 0.5 and 2 nm to a total Q interval of 10−2–5 nm−1. The resolution and the sample detector distance LD can be adjusted in choosing the length of collimation LS. The space between selector and first aperture is bridged by neutron guides which, in segments of 1 m length, can be moved in or out of the beam; in this way at the expense of resolution, the neutron intensity at the sample can be remark­ ably enhanced by a beam of larger divergence. This is shown in Figure 3 where the measured primary intensity for various wavelengths is depicted versus the length of collimation. The intensity is inversely proportional to the square of the colli­ mation length. Only for 0.45 nm, wavelength deviations are observed at 2 m because of the limited divergence emitted from the neutron guide. The optimal condition of the instrument with respect to resolution and intensity is achieved, when all elements of resolution in eqn [8] contribute the same amount to the reso­ lution broadening. From this, the following instrumental 2dS. The corresponding setting results: LD = LS and dE = d�D =pffiffiffi� resolution according to δQopt: ¼ k= 3 ⋅ dE =L (eqns [8] and [7]) leads to an intensity at the sample according to9 ΔI0 ¼ L ΔΩ

ΔΩD, Δλ/λ

Δk/k

k

LD = 1–20 m

½5

� � Φ k 4 −ðk=kT Þ2 Δk e k 2π kT

L¼

LC = 1–20 m

ΔΩ Q ΔΩD

Figure 1 Traces of neutrons in real and reciprocal space.

� �4 F 2 δQ LD ¼ L L2D 2 k LD

½9

The relationship shows that the intensity at the sample is proportional to the square of the instrumental length and is the reason for the typically 40-m-long-pinhole SANS instru­ ments. The upper intensity limit is determined by the maximum divergence of the neutrons transmitted by the neu­ tron guide according to the total angle of reflection.

Structure Characterization in Fourier Space | Neutron Scattering

Neutron flux (cm–2 s–1)

109

100

λ= 0.45 nm 0.7 nm 1.2 nm

108

10–1

1.96 nm

Normalized intensity

334

107 106

L–2

105

10–2 10–3 10–4 10–5

104

1

10 Collimation length (m)

2Γ1/2

10–6

Figure 3 KWS 2 at FRM II: Neutron flux at sample for different length of collimation and neutron wavelength (δλ/λ = 0.2 and entrance aperture 3  3 cm2). The inner area of the neutron guides is 5  5 cm2 and are coated with a nonmagnetic Ni–Ti/Mo supermirror with a critical angle of m = 1.25 similar to Ni58.

A further important criterion for the quality of an smallangle scattering (SAS) diffractometer is the sharpness of the primary neutron beam prepared by the collimator. In Figure 4, the primary beam for a given configuration is shown in a semi-logarithmic presentation. One recognizes that at twice the full half width the primary intensity has decreased to values between 10−5 and 10−6. This result demon­ strates that the instrument is in good condition, for example, the background near the primary beam is sufficiently small, so that also in this region neutrons scattered from the sample can be sensitively detected and analyzed. In order to study the total Q range of pinhole SANS between 10−2 and 5 nm−1, one needs several discrete settings of the detector. For ‘older’ detectors with about 70 cm diameter, three settings are needed. Presently, larger detectors with a neutron-sensitive area of the order of 1 m2 are available that only need two settings. These detectors simultaneously allow a larger Q interval to be studied, which might also be relevant for real-time experiments. On the other hand, the adjustment of the instrument with respect to resolution and primary intensity might not be optimal. Another relevant extension of the pinhole SANS technique is the focusing of a small source aperture by refractive lenses in front of the sample onto high-resolution detector.10,11 Aspherical lenses of MgF2 are the latest development permitting 11 m

–20 –15 –10

2Γ1/2

–5 0 5 Position (cm)

10

a sample area of 5  5 cm2. In this way, using large samples, 1 order of magnitude in intensity may be gain or focusing on a small beam spot 1 order of magnitude smaller Q of the order of 10−3 nm−1 may be achieved. Both SANS instruments, KWS-1 and KWS-2, at the FRM II in Garching are equipped with MgF2 aspherical lenses. They are arranged in a separate housing of the collimator in front of the sample and may be cooled to 70 K, in order to avoid thermal diffuse scattering thereby enhan­ cing the transmission of the lenses. Chromatic aberration may be avoided in narrowing δλ/〈λ〉 by a chopper that is installed in both instruments. A detailed discussion of aspherical MF2 neu­ tron lenses is given in Reference 11.

2.11.2.1.3

Toroidal mirror focusing SANS

The design of a SANS instrument using a toroidal mirror as focusing element is depicted in Figure 5 and has been realized for the KWS-3 diffractometer at the FRM II in Garching. A detailed description of the principles of this instrument can be found in References 9 and 12 Monochromatic neutrons enter the instrument through the entrance aperture with a diameter of about 0.1 cm. The neu­ trons are reflected at a toroidal mirror (and replica mirrors, future option), and are detected at the focal point by a highposition-resolution detector. The toroidal (replica) mirror has an area of 20  120 (20  60) cm2 and is coated with 100 nm 11 m Detectorposition reflectometer

Replica mirror Mirror chamber

Position-sensitive detector

Sample

1.20 m

Figure 5 Design of a focusing SANS.

20

Figure 4 Resolution curve for a given configuration.

Toroidal mirror

Entrance aperture

15

L,10 m

Detectorposition small-angle scattering

Structure Characterization in Fourier Space | Neutron Scattering of 65Cu. This instrument allows small-angle scattering as well as reflection experiments. The position of the sample is just behind the mirrors. The instrument has the total length of 22 m and mirrors of 1.2 m length in the middle between aperture and detector. Neutrons of 1.5 nm wavelength cover a Q range of 10−2–10−3nm−1. The intensity at the sample is given as �

F δQ ΔI0 ¼ L ΔΩ 2 L2D ¼ L ð4γ2C Þ k LD

Sample

Optical bench

½10

At these small Q, this instrument is superior to the pinhole instrument because the instrumental resolution is decoupled from the divergence of the primary neutron beam, that is, the full solid angle 4γ2C of the neutron guide can in principle be used. Of course, the available total area of the mirrors as well as the sample size area limits the full use of that intensity. The concept of the focusing SANS has been known for a long time.9 However, only recently has it been possible to build such instruments with the necessary good quality of the primary beam profile as shown from the experimental resolution curve shown in Figure 6. The reason is the high demand on surface quality of the mirror; sufficiently high quality mirrors can be built today as a result of an extended project for the development of X-ray satellites.

2.11.2.1.4

Channel crystals

�2 L2D

Double crystal diffractometer

Out of all SANS instruments, the DCD allows the smallest Q to be measured. Its concept is presented in Figure 7. Two perfect silicon single crystals mounted on an optical bench are the central part of this instrument. The reflec­ tivity of perfect crystals is described by the Darwin curve according to9 � 1 jyj≤1 RðyÞ ¼ ½11 1−ð1 − y−2 Þ1=2 jyj >1 The parameter y is a reduced scattering angle and the reflectivity R is equal to 1 within the interval of |y| ≤ 1 with y ¼ ðθ−θB Þ=Δθ (θB the Bragg angle). The width of total reflection is determined as

335

Detectors

Figure 7 Schematic design of a double crystal diffractometer.

Δθ ¼

be−W jFjNλ2 4π sin θB

½12

where b is the coherent scattering length, e−W the Debye–Waller factor, F the geometric structure factor, N the particle density, and λ the neutron wave length. For instance, for the (331) lattice planes of a silicon single crystal and λ = 0.18 nm neu­ trons the Bragg angle of diffraction is about 45 °. Then the half width of the resolution curve is Δθ ¼ 3:2 μrad, which corre­ sponds to an angle of slightly more than half a second of arc or a Q ¼ 1:12  10−4 nm−1 . For experiments, the second crystal is rotated with respect to the first one. When the corresponding lattice planes of both crystals are oriented parallel to each other, the Darwin curves of both crystals overlap completely and the maximum intensity is transmitted (Figure 8 at θ = 0). For finite rotation angles, both Darwin curves only partly overlap with the result of a smaller reflected intensity. The rotation of the second crystal is mathematically equivalent with the folding of the two Darwin curves, which gives the s–s mode (single–single bounce) resolution curve in Figure 8. This method demands very precise angular settings and therefore needs protection against mechanical vibrations, fluctuations of temperature, and so fourth. The advantages of this instrument are its very high resolu­ tion and its relatively simple and low cost design in comparison with most other neutron scattering instruments.

100

101 Si111; λ = 4.48 Å

100 Intensity I(θ)/I(0)

I(Q)/I(Q = 0)

10–1

10–2

10–3

10–1 10–2

t–t mode

10

10–5 –4

–9

–6

–3

0 Q (10−3 nm−1)

Figure 6 Resolution curve of a focusing SANS.

3

6

9

s–s mode

–3

10–4

ΔQ0.5=3.85×10–4Å−1

10

2 Δθ (t–t)

Δθ(s–s) = 54.2 μrad Δθ(t–t) = 25.8 μrad

–90

–60 –30 0 30 60 Scattering angle θ (μrad)

90

Figure 8 Experimental resolution after single–single and triple–triple reflections within the channel-cut crystal.

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Structure Characterization in Fourier Space | Neutron Scattering

labeling enables highlighting of different constituents of the structure. NSE spectroscopy allows analyzing of the quasielastic energy broadening that the neutrons suffer upon scattering at the sample. This pertains the intensity that is measured by SANS without energy analysis. Since in polymeric systems the quasielastic linewidths are many orders of magnitude smaller than the energy of the incoming neutrons, a special technique – NSE spectroscopy16 – has to be used to achieve the required resolution. However, NSE spectroscopy does not yield the qua­ sielastic spectrum but rather its Fourier transform, the intermediate scattering function. In terms of scattering length density, this is (see also eqn [4]) � � ½13 SðQ; tÞ ¼ ρQ ðtÞρ−Q ð0Þ Figure 9 Three bench channel-cut Si perfect crystal. The cut between the first and third area of bench strongly improved the background conditions.

Disadvantages are that it measures in slit geometry and that the experimental points are measured in sequence. There are, how­ ever, improvements possible by special designs of the crystals. The relative high background of the s–s mode near the primary beam can be strongly reduced by so-called channel-cut crys­ tals.13 If the neutrons are reflected within the channels of a compact single crystal (see photo in Figure 9), for example, 3 times (three bounce mode), then the Darwin curves of the two crystals are multiplied 3 times with themselves and the folded reflection curve appears as the triple–triple (t–t) mode (red points in Figure 9). An important achievement of the high quality experimental t–t mode resolution curve was the addi­ tion of a groove between the area of first and third bounce area as depicted in Figure 9.14 Figure 10 shows an example of combining all three SANS methods covering a Q range of 3.5 orders of magnitude.15 Spherulites of crystallized syndio-Polypropylene in C10D22 of Rg = 3.3 μm radius of gyration and their inner structure have been detected.

2.11.2.2

Spin Echo Spectroscopy

Neutron diffraction – in particular SANS – yields information on the microscopic structure of the sample, proper use of H/D

Thus, SANS yields S(Q, 0), which corresponds to the average of instantaneous structures, whereas NSE spectroscopy yields additional information on the temporal variation of the struc­ tures that contribute to the scattering intensity. The motions observed are thermal fluctuations, their correlation time depends on a balance of friction, and eventually present forces restoring equilibrium configurations. If no restoring forces act, the observed Brownian motion is diffusion without bounds, if microscopic confinement by external potentials or topological effects is present, the diffusion is bounded and the correlation function halts at a finite level. The NSE technique relies on the use of the neutron spin as an individual timekeeper for each neutron. The schematics are shown in Figure 11. In a preparation zone (precession 1), for example, a magnetic field from a solenoid, the timekeeping starts by the action of the first π/2-flipper, which rotates the neutron spins by 90° such that they are perpendicular to the precession field. The time of flight of the neutrons is coded in terms of Larmor precession angles accumulated on the way to the π-flipper, close to the sample. The flipping operation (spin rotation by π = 180° around an axis perpendicular to the pre­ cession axis) effectively inverts the Larmor precession angle. After scattering at the sample a symmetric precession track add further spin rotations such that a neutron without velocity change at the sample regains its starting value. That is, the initial polarization at the position of the second π/2-flipper is restored. This effect is called ‘spin echo’ and is analogous to the

(b)

(a)

dΣ/dΩ (cm–1)

108 106 Rg = 3.3 μm

104

–2

Q

102 100 10–2 –4 10

Q

–4

0.5% s-PP in d-22 room temp.

10–3

10–2 10–1 Q (nm–1)

100

Figure 10 (a) Scattering pattern as determined from DCD, focusing, and pinhole SANS instruments. Crystallization of syndio-Polypropylene in d-22 solution – large scale view covering 3.5 orders of magnitude in Q and 10 orders of magnitude in intensity. (b) Optical micrograph of the spherulitic morphology. From Radulescu, A.; Kentzinger, E.; Stellbrink, J.; et al. Neutron News 2005, 16 (2), 18–21; Radulescu, A.; Schwahn, D.; Stellbrink, J.; et al. Macromolecules 2006, 39, 6142–6151.15

Structure Characterization in Fourier Space | Neutron Scattering

(a)

Neutron spin

Spin rotation

π/2

π

intensities with and without spin flip without Larmor analysis, respectively. In the case of a sample that scatters both coher­ ently with Icoh and incoherently with Iinc in a calculation analogous to eqn [15], we need to consider that with a prob­ ability of 2/3 in spin coherent scattering the neutron spin is flipped. With that, eqn [15] transforms to

Δλ /λ = 10% π/2

magnetic field

(b)

π Sample π/2 Precession 1 Flipper Precession 2 π/2

Detector analyzer

~ S NSE ðQ; tÞ ¼

π (c) Spin echo (d)

π

Pmax P(τ) Reduced spin echo

Figure 11 Schematics of a generic (IN11 type) neutron spin echo (NSE) instrument. (a) Shows the development of the spin vector during neutron passage through the instrument. (b) Indicates a possible magnetic coil arrangement corresponding to the J-NSE instrument in Garching and the NSE instrument at NIST. (c) Indicates the projected spin directions at the start and end points of the neutron preparation and analysis flight paths from π/2 to π and from π to π/2 flipper in the case of elastic scattering, and (d) Indicates the situation in the case of quasielastic scattering.

Hahn echo known from NMR.17 After the π/2-flipper, the polarization in z-direction is again rotated in the direction of the magnetic guide field and then probed by the analyzer which transmits neutrons to the detector with a probability that ideally is T ¼ ½1 þ cosðΨÞ=2, with Ψ the resulting preces­ sion angle at the π/2 flipper. The effect – as illustrated in Figure 11(c) – is independent of the initial neutron velocity thus allowing a broad (10–20%) width of the incoming neutron wavelength distribution with corresponding high intensity. If the neutron changes its velocity during scattering the final polarization is reduced (Figure 11(d)). Integration over a quasielastic spectrum yields a detected beam polarization corresponding to the Fourier transform of the spectrum, that is, the (normalized) intermedi­ ate scattering function. Ideally, the detected intensity after polarization analysis is 2 3 ! 7 6 ð 2 7 1 6 6SðQÞ  cos Jλ3 γ mn ω SðQ; ωÞdω7 ½14 IðQ; tÞ ¼ 6 7 2 4 2π h2 5 |fflfflfflfflfflffl{zfflfflfflfflfflffl} t

where the effective magnetic field integral J can easily be con­ trolled by the electrical current in the solenoids and thus be used to scan the time parameter t, that also depends on the cube of the neutron wavelength λ. For a purely coherent scatter­ ing sample, eqn [14] may be easily rewritten in terms of the observed final neutron polarization Iþ ðQ; tÞ − I− ðQ; tÞ SðQ; tÞ ¼ SðQÞ I↑ − I↓

337

½15

where I+ and I− are the intensities for the positive and negative echo amplitudes and I↑ and I↓ are the transmitted

Icoh

Scoh ðQ; tÞ 1 − Iinc Sinc ðQ; tÞ SðQÞ 3 Icoh − 31 Iinc

½16

~ NSE ðQ; tÞ is the effective time-dependent polarization Thereby S detected at the instrument. We note that the conversion of the ‘ideal’ intensities to S(Q, t) under real experimental conditions need to consider polarization losses in the instrument that are measured in a reference experiment on a purely elastic scatterer. Such a measurement serves as a determination of the experi­ mental resolution function. Since the experiments are performed in Fourier space instead of a deconvolution as required in ω-space, the resolution corrections merely requires a division of the sample data by the reference measurement. In the appendix, we present a more detailed description of the NSE instrument including its mathematical analysis. At the time when this chapter was written, high-resolution NSE instruments are operated by the ILL, Grenoble (http://www. ill.eu), JCNS at the FRM II reactor in Garching (http://www.jcns. de, http://www.frm2.tum.de), the NIST in Gaitherburg (http://www.nist.gov/instruments/nse), the SNS at Oak Ridge (http://www.sns.gov, http://www.jcns.de), and the JRR-3 in Tokai (http://www.issp.u-tokyo.ac.jp/labs/neutron/inst/NSE/ iNSE_Index.html). A peculiarity of the standard NSE technique is that the spin-incoherent scattering contribution undergoes a spin flip in 2/3 of the scattering events. Then the echo amplitude corre­ sponds to –1/3 of the intensity and 2/3 go into an additional unpolarized background. In the typical SANS regime with appropriate H/D contrast in the sample, the corresponding coherent scattering dominates up to Q = 2–3 nm−1 beyond that value incoherent scattering (background of the SANS data) is larger. In the overlap region, a complicated compound signal of coherent and –1/3 of incoherent dynamics may com­ plicate the interpretation (see eqn [16]). On the other hand, a fully protonated sample will dominantly scatter (spin)-incoherently and thus allow for the observation of the proton (i.e., segmental/molecular) self-correlation.18

2.11.3 Representative SANS Results In this section, the theoretical basis of SANS will be exposed as is needed in polymer physics and will be clarified with simple experimental examples. Since the end of 1939, the method of SAS was developed by Guinier and Kratky and was mainly applied for questions in metal physics.19,20 In one of his first experiments, Guinier correctly interpreted scattering from cop­ per precipitates in aluminum, the so-called Guinier–Preston zones. Today, SANS techniques are broadly used in soft matter; a main reason is the relatively simple possibilities of contrast generation and variation for polymer chains and other hydro­ carbon molecular items by the exchange of hydrogen and deuterium.

338

2.11.3.1

Structure Characterization in Fourier Space | Neutron Scattering

Polymer Chain Conformation

As a model of a linear chain, we consider a polygon of z vectors of length jr i j, whose directions are randomly distributed in 3D. Such a polygon of a freely joint chain is shown in Figure 12. Each vector r i represents a monomer, the chemical unit of a polymer of length b. On a coarse-grained scale, such a chain can also be described by another polygon with larger vectors r i representing several monomers. Both polygons correctly repre­ sent the global properties of the chain as the end-to-end vector R, evaluated from the sum of all vectors according to R¼

z X

ri ¼

i

~z X

r i

½17

i

The probability density that a freely joint chain with N mono­ mers takes a conformation with the end-to-end vector R can be approximated by the Gaussian distribution according to �

3 2π Nb2

WðR; NÞ ¼

�3=2

� � 3R2 exp − 2Nb2

½18

The statistical average of the mean end-to-end distance is zero (〈R〉 = 0), whereas its mean square deviation is linearly propor­ tional to the number of monomers Nð〈R2 〉 ¼ R2L ¼ b2 N Þ. The radius of gyration is given as R2g ¼ RL2 =6. Equation [18] is the result of a ‘central limit theorem’ from elementary probability theory.21 A linear ‘real’ chain is different from a freely joint chain in so far as neighboring monomers are correlated. This correlation is considered by the parameter cN in R2g ¼ cN b2 N=6 or expressed pffiffiffiffiffi by the statistical segment length according to l ¼ b cN and determined from radius of gyration according to R2g ¼ l2 N=6. So, a statistical segment length of polystyrene (PS) is r1*

r1

Rij

R

Ri Rj Figure 12 Model of a freely joint linear chain.

(a) 2.2

determined as lPS = 6.8 Å. A realistic chain can therefore be represented by a polygon of N vectors of segment length l. The form factor P(Q) of a linear chain measured in a SANS experiment is determined according to eqn [2] as the sum of the phase factors of the monomers averaged over all chain configurations P ðQÞ ¼

P ðQÞ ¼

� � N b2 2 1 X exp −ji − j j Q 6 N 2 ij

½20

After approximating the sums by integrals and by some calcu­ lation, the well-known Debye formula is achieved PDb ðQÞ ¼

2 ½x−1 þ expð−xÞ x2

½21

with x ¼ Rg2 Q2 . It is instructive to consider the Debye form factor in the regions of small and large Q compared with the inverse size of the polymer; these approximate form factors have a much simpler form and can easily be used for the analysis of the scattering data. So in the region of small Q, for example, Q  1/Rg one finds PDb ≅ 1− R2g Q2=3. This Zimm approxima­ −1 ≅ 1 þ R2g Q2 =3 is displayed in Figure 13(a). From this tion PDb plot, the radius of gyration is easily determined. In the region of large Q(Q  1/Rg), the expression PDb ≅ 2=ðQ ⋅ Rg Þ2 is obtained. As shown in Figure 13(b), it follows a power law with a slope of –2 consistent with the D = 2 fractal dimensionality of poly­ mer chains with a Gaussian conformation.

(b)

2.0 Debye

1.6 1.4 Zimm

1.2

(QRg)2 P(Q)

P –1(Q)

½19

The meaning of the vector R i becomes clear from Figure 12. In a macroscopically large sample, the number of polymers is suffi­ ciently large, to describe the polymer conformation with the probability distribution of eqn [18]. The average value of the phase factors between the positions i and j within the chain follows from the approximate Gaussian distance probability eqn [18] of any section of the freely jointed chain: ð � � � � � � exp i Q R ij ¼ d3 R ij w Rij ji − jj exp iQR ij ¼ Wij ðQÞ with � � 2 Wij ðQÞ ¼ exp −ji − jj b6 Q2 . For the form factor, we get

2.0 1.8

N � � 1 X exp i Q ðR i − R j Þ 2 N ij

1.5 1.0 0.5

1.0 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (Q Rg)2

0.0 0

2

4

6

8

10

RgQ

Figure 13 (a) Zimm representation: within the Zimm approximation the plot of 1/P versus Q2 gives a straight line. The slope of the straight line is proportional to Rg. (b) ‘Kratky plot’ of Debye’s form factor: at large Q the representation of Q2P(Q) versus Q gives a constant value.

339

Structure Characterization in Fourier Space | Neutron Scattering

2.11.3.1.1

Homopolymer melts

We now consider a melt of n linear polymer chains of molar volume Vchain. The polymers are of the same type and consist of N monomers with the coherent scattering length b. In total there are Ntot = nN monomers with n the number of chains in the sample of volume VS. The scattering cross section is derived from eqns [1] and [2] and averaged over all chain conforma­ tions according to dΣ b2 ¼ dΩ Vs

*� �+2 Ntot Ntot �X � � iQR � b2 X iQR i � � e ¼ e ij � � V s ij i

½22

In a next step, eqn [22] is decomposed into ‘intramolecular’ and ‘intermolecular’ interference terms P(Q) und W(Q) according to 2 3 6 X 7 N X 7 6 N dΣ b2 6 ¼ 6n < eiQRij > þ nðn−1Þ < eiQRij >7 7 |fflfflffl{zfflfflffl} ij dΩ Vs 4 ij 5 ≅n2 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} N 2 PðQÞ

which at any finite Q does not contribute. Thus, for any finite Q, we have dΣ 1 ðQÞ ¼ Δb2 SDD ðQÞ dΩ Vs

½29

where SDD describes the intra- and intermolecular interferences between all deuterated monomers. The definition of the occupation operators in eqn [26] implies the condition of incompressibility, as no free volume is considered. Following eqn [23], the partial structure factor of the deuterated monomers is given SDD ðQÞ ¼ Φ n N 2 PðQÞ þ Φ2 n2 N 2 WðQÞ

½30

where Φ is the volume fraction of the deuterated polymer component. Because of the Babinet principal, we get P(Q) = –nW(Q) and dΣ Δρ2 ðQÞ ¼ Φð1−ΦÞVchain PðQÞ dΩ NA |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |ffl{zffl} SðQÞ

½23

½31

K

N 2 WðQÞ

with the structure factor

and

SðQÞ ¼ Φ ð1− ΦÞ Vchain PDb ðQÞ ½24

where ρ = b/V0 is the coherent scattering length density, Vchain the molar volume of the chain, NA the Avogadro number, and V0 the molar volume of the monomers. Incompressible melts of identical polymer chains show no thermal density fluctuations and therefore no diffraction occurs in the Q range of SANS. Therefore, the scattering cross section of eqn [24] must be zero and the following relation­ ship holds P ðQÞ ¼ −nWðQÞ

½25

representing the Babinet principal.

2.11.3.1.2 Melt of deuterated and protonated polymers with chemically identical monomers of same segment length We now consider a melt of chemically identical polymers whose components are either protonated or deuterated. For further consideration, we introduce an occupation operator σi � 1 bi ¼ bD σi ¼ ½26 0 bi ¼ bH The coherent scattering length of a monomer at the position i is then described as bi ¼ σ i ðbD − bH Þ þ bH ¼ σ i ⋅ Δb þ bH

½27

Substituting in eqn [2], this expression yields dΣ 1 ðQÞ ¼ dΩ Vs

�2 + *� Ntot � Ntot � X � X iQ R i iQRi � σi e þ bH e �Δb � � i¼1 � i¼1 |fflfflfflfflffl{zfflfflfflfflffl}

½28

≅ δðQÞ

where, in the right-hand side, the first part resembles the scat­ tering amplitude from all the deuterated monomers. The second part as a sum over all monomers gives a δ function,

½32

Thus, the change of contrast of chemically identical polymers by isotope substitution enables the experimental determination of the form factor of a single chain in a melt. In Figure 14, an experimental example is shown for a polystyrene melt confirming the –2 power law at large Q as predicted by the Debye form factor.

2.11.3.1.3

H/D polymer blend in solution

Next, we derive the cross section of an isotopic polymer blend in solution. Again, we introduce occupation operators � 1 bi ¼ bD σ D ðR i Þ ¼ ½33 0 bi ¼ bH or bO � 1 bi ¼ bH σ H ðRi Þ ¼ 0 bi ¼ bD or b0

d −PS/PS S (Q)/φ(1−φ) (cm3 mol–1)

� dΣ ρ2 � ðQÞ ¼ Vchain PðQÞ þ nWðQÞ dΩ NA

105

φ =0 . 4 8

VW = 0.91×106 cm3 mol–1

104

Q−(2.01±0.002)

103

102 10–3

10–2

10–1

Q (Å–2) Figure 14 Structure factor of a 50% mixture of deuterated and proto­ nated polystyrene in double logarithmic representation. The power law behavior at large Q is described by statistical chain.

340

Structure Characterization in Fourier Space | Neutron Scattering

where b0 is the scattering length of the solvent. The scattering cross section becomes *� Ntot � dΣ 1 ��X ðQÞ ¼ bD σ D ðr i Þ þ bH σ H ðr i Þ � dΩ Vs � i¼1 � + �2 � �� iQ R i� e þb0 1− σ D ðr i Þ − σ H ðr i Þ � �

Block H

Block D Figure 15 Presentation of a diblock copolymer.

which after some calculation leads to the form * " Ntot dΣ 1 X ðQÞ ¼ ðbD − b0 Þσ D ðr i Þ þ ðbH − b0 Þσ H ðr i Þ dΩ Vs i¼1 � + # � iQ R i� ½34 þ b0 e �2 |ffl{zffl} � ≅0

and, finally, to a sum of partial structure factors weighted with corresponding contrast factors ΔbD = bD – b0 and ΔbH = bH – b0 � dΣ 1� 2 ðQÞ ¼ ΔbD SDD ðQÞ þ 2ΔbD Δ bH SDH ðQÞ þ Δb2H SHH ðQÞ dΩ Vs ½35 The partial structure factors are given as SDD ðQÞ ¼ ΦnN 2 PðQÞ þ Φ2 n2 N 2 WðQÞ

½36

2

SHH ðQÞ ¼ ð1− ΦÞnN PðQÞ þ Φ n N WðQÞ

½37

SDH ðQÞ ¼ Φð1− ΦÞn2 N 2 WðQÞ

½38

2

2 2

Using these, the following scattering cross section is derived dΣ 1n ðQÞ ¼ ðbD − bH Þ2 Φ ð1− ΦÞ n N 2 PðQÞ dΩ Vs � �o þðb Poly − b0 Þ2 n N 2 PðQÞ þ n2 N 2 WðQÞ

½39

For this system, the Babinet principle is not valid as it contains polymers and solvent molecules. If one matches the scattering length of the solvents and � � the averaged one of the polymers b Poly ¼ ΦbD þ ð1−ΦÞ bH according to b Poly ¼ b0 (‘zero’ con­ trast), the second term of eqn [39] does not contribute to the scattering and, consequently, the single-chain form factor can be determined according to dΣ ðQÞ ¼ c ⋅ Φ ð1− ΦÞ Vchain PðQÞ K dΩ

½40

where c is the volume fraction of the polymers in solution, Φ and (1–Φ) the volume fraction of the deuterated and proto­ nated chains, respectively, and K the contrast factor.

2.11.3.1.4

Scattering from diblock copolymer melts

We now derive the scattering cross section of a linear polymer, consisting of the two blocks ‘A’ and ‘B’ of different monomers that are symmetric with respect to the number of the monomers N = 2ND = 2NH (Figure 15). The scattering cross section is given as dΣ 1 ðQÞ ¼ Δb2 SDD ðQÞ dΩ Vs

½41

with the relationship SDD = SHH = SDH because we assume an incompressible melt. The various partial structure factors are given as

� �2 � �2 N N PDD ðQÞ þ n2 WDD ðQÞ SDD ðQÞ ¼ n 2 2 � �2 � �2 N N SHH ðQÞ ¼ n PHH ðQÞ þ n2 WHH ðQÞ 2 2 � �2 � �2 N N SDH ðQÞ ¼ n PDH ðQÞ þ n2 WDH ðQÞ 2 2

½42 ½43 ½44

The difference of the diblock partial structure factors with respect to the corresponding ones of the binary blend in eqns [36]–[38] is the PDH(Q) form factor in eqn [44]. In order to express the partial structure factors only by intramolecular interferences, the form factor of the total chain PT(Q) according to PT ðQÞ ¼

PDD ðQÞ þ PDH ðQÞ 2

½45

is needed. Because the intermolecular interference terms of the chain segments as well as of the total chains according to WT = WDD = WHH = WDH are the same in combination with the relationship PT ðQÞ ¼ −nWT ðQÞ according to the Babinet principle (eqn [25]), we finally obtain the partial structure factor for block D � �2 � N � SDD ðQÞ ¼ n PDD ðQÞ−PT ðQÞ 2

½46

and the scattering cross section � dΣ Vchain � ðQÞ ¼ K PDD ðQÞ−PT ðQÞ dΩ 4

½47

It is the difference of the intramolecular form factor of a single block (PDD, PHH, and PDH) and of the total chain (PT). These form factors of a symmetrical diblock copolymer are plotted in Figure 16. An interference peak is observed whose position Q * Rg = 1.9 is related to the radius of gyration. For Rg = 10 nm, Q* is observed at 0.19 nm−1. The observation of an interference peak in diblock copolymers is plausible from the consideration that composition fluctuations of the blocks A and B can only occur on the length scale of the polymer. An experimental example is shown in Figure 17.

2.11.3.2 Binary Homopolymer Blends: Random Phase Approximation The structure factor of an ideal binary polymer melt is evalu­ ated within the random phase approximation (RPA) according to22,23

Structure Characterization in Fourier Space | Neutron Scattering

1.0

0.20 Rg = 10 nm

P

0.5

Diblock copolymer

0.15

PDD

Rg = 10 nm

Q*

P(Q)

P(Q)

341

0.0 –0.5 –1.0 0.0

0.05

–PT

0.1

0.2 Q

0.10

0.3

0.4

0.5

0.00 0.0

0.2

(nm–1)

0.4 0.6 Q (nm–1)

0.8

1.0

Figure 16 Form factor of a symmetrical diblock copolymer.

S(Q) (103 cm3 mol–1)

10

well as no phase transition phenomena. Such ideal polymer solutions are not found in reality; as demonstrated in Figure 18, even isotopic mixtures of chemically identical poly­ mers show phase separation at low temperatures because of small but finite mixing interaction energy. For polymer blends, such interaction is described by the FH parameter

d-PB(1,4)-PS T = 100 °C

8 6

Γ¼

4

0.2

0.4 0.6 Q (nm–1)

0.8

S−1 ðQÞ ¼ S−1 0 ðQÞ − 2Γ

Figure 17 Structure factor of a PB/PS diblock copolymer.

S−1 0 ðQÞ ¼

½49

The FH parameter has the meaning of a free enthalpy of mixing with the enthalpic and entropic terms Γh and Γσ, respectively. In RPA approximation, the interaction parameter is added to the structure factor S0(Q) according to

2 0

Γh − Γ σ T

½50

For small QRg, the structure factor can be written in ‘Zimm approximation’ as

1 1 þ B ðQÞ A ðQÞ ð1− ΦÞVB PDb ΦVA PDb

S−1 ðQÞ ¼ S−1 ð0Þ þ AQ2

½48

½51

−1

The inverse structure factor is determined from the sum of the inverse Debye form factors of both chains weighted with their molar volumes and volume fractions. For identical molar volume (VA = VB) and form factor PDb(Q), we get the structure factor of eqn [32]. Equation [48] corresponds to an ideal solu­ tion of two components whose mixing energy is zero and therefore there are no thermal composition fluctuations as

with the inverse structure factor S (0) = 2[Γs – Γ] and the FH 1 parameter at the spinodal temperature, 2 ΓS ¼ ΦV1 A þ ð1− ΦÞV , Γs B is inversely proportional to both chain molar volumes and is related to the translatorial entropy of mixing. The parameter A is composed of the square of the statistical segment lengths of both chains as discussed in context with eqn [19]. Experiments, in general, are performed at sufficiently small Q in order to use the simple form of eqn [51] for analysis of the scattering data.

500

Temperature (˚C)

450

d-PS/PS Vw = 1.42 106cm3 mol–1

(Γ = 0; TComp)

400 350 300

Critical point: ΦC;TC

ΓC = 2/VW; TC

250 200 150 0.0

Binodal

0.2

Spinodal

0.4

0.6

Composition

0.8

0.0

0.5

1.0

1.5

2.0

2.5

Γ (10−6mol cm−6)

Figure 18 Phase diagram and Flory–Huggins interaction parameter of a symmetric (VD = VH) isotopic dPS/PS blend of 1.42  106 cm3 mol−1 molar volume.

Structure Characterization in Fourier Space | Neutron Scattering

As shown in Figure 18, the spinodal temperature represents the phase boundary of a symmetric blend (molar volume Vchain = VD = VH) between metastable and unstable two-phase regions. At the critical point, the unstable region touches the stable one-phase region. In the homogeneous phase at high temperatures, the FH parameter is smaller than ΓC = 2/Vchain in accordance with a positive S(0) fulfilling the Gibbs condition of stability; S(0) represents a susceptibility which, following the fluctuation-dissipation theorem, is related with the free enthalpy of mixing ΔG according to

½53

The first two terms describe the combinatorial entropy of mix­ ing, while the last term represents a ‘segmental’ free energy of mixing involving the FH parameter χ. Originally, χ was defined as being proportional to an enthalpic term χh according to χ = χh/T; later this parameter had to be extended to a free energy of mixing χ = χh/T – χσ by introducing an empirical noncombi­ natorial entropic term χσ. The necessary extension of the FH parameter to a free energy of mixing was noted by Flory24 and later forced mainly by SANS experiments. The combinatorial entropic part is inversely proportional to the polymer molar volumes VA and VB. Thus, the phase boundaries of many polymer blends can easily be shifted into an experimentally accessible temperature range by a proper choice of the molar volume (i.e., the molecular weights, respectively their lengths). RPA (eqn [50]) and FH theory (eqn [53]) give the same result for S(0) as both theories assume mean-field approxima­ tion. Only the FH parameter might be different because of its dependence on composition, for example, χ = χ(Φ) and the relationship of susceptibility and thermodynamic potential in eqn [52] leading to 2Γ ¼ ∂2 ½Φð1− ΦÞχ=∂Φ2 . We therefore distinguish the FH parameter determined from SANS with the symbol Γ from the original symbol χ in eqn [53] as eqn [52] is the more basic equation. In case of Γ > ΓC the susceptibility S(0) becomes negative. The system becomes unstable with respect to long wavelength fluctuations and decomposes following the mechanism of spinodal decomposition in two macroscopically large phases where one polymer component is dominating.

2.11.3.2.1

Phase separation and miscibility

The time-dependent structure factor of the early stages of spi­ nodal decomposition is described by the Cahn–Hilliard–Cook mean-field theory25 � � SðQ; tÞ ¼ STf ðQÞ þ ST0 ðQÞ − STf ðQÞ L2 ðQ; tÞ ½54 This relation represents a relaxation process between the two equilibrium structure factors ST0 ðQÞ and STf ðQÞ at T0 and Tf in the one- and two-phase region, respectively. The factor L(Q, t) represents the normalized dynamical structure factor which describes the time evolution of this process.26 In the

½55

is related to the collective diffusion constant DC(Q) = D0(1–Γf/ΓC) (D0 self-diffusion constant and Γf the FH parameter at the temperature of phase decomposition Tf) and the Onsager trans­ port coefficient Λ. For polymers, the Onsager coefficient is a nonlocal quantity of the form ΛðQ ⋅ RΛ Þ ¼

½52

ΔG is normalized by the gas constant R and absolute tempera­ ture T. The free enthalpy of mixing ΔG of polymer blends was originally formulated within the mean-field approximation for incompressible polymer blends by Flory and Huggins as ΔG Φ ð1− ΦÞ ¼ ln Φ þ lnð1− ΦÞ þ Φð1− ΦÞ χ VB RT VA

~ RðQÞ ¼ Q2 DC ðQÞ ¼ Q2 ΛðQ ⋅ RΛ Þ S−Tf1 ðQÞ

Λ0 ½1− expð−R2Λ Q2 Þ Q2 R2Λ

½56

with an interaction range RΛ of the same length as the radius of gyration.27 In Figure 19, the phase diagram of a symmetric isotope mixture deuterated PS (dPS)/PS of molar volume of 1.42  106 cm3 mol−1 is depicted as determined from SANS experiments.28 Samples of critical composition of ΦC = 0.5 were equilibrated at 240 °C (solid symbol) and then within the spi­ nodal region heat treated at various temperatures (open symbols) in order to follow the early stages of spinodal decomposition. The evolution of phase decomposition is shown in Figure 20 at T = 170 °C by means of the normalized time­ 300 One-phase region

Temperature (°C)

∂2 ðΔG=RTÞ S−1 ð0Þ ¼ ∂Φ2

Cahn–Hilliard–Cook theory, L(Q, t) assumes a single exponen­ ~ ðQÞt. The relaxation rate tial form according to LðQ; tÞ ¼ exp½−R

d-PS/PS Vw = 1.42 106cm3 mol−1

250

Critical point Binodal

200 Spinodal

150 Two-phase region

100 0.0

0.2

0.4

0.6

1.0

0.8

Composition Figure 19 Phase diagram of dPS/PS blend.

6 T = 170 °C 2h 4h 8h 22 h 70 h 238 h

5

4 σ (Q, t)

342

3

2

1

0

1

2

4 5 3 Q (10−2 nm−1)

6

7

Figure 20 Time-dependent structure factor of the early stage of spinodal decomposition. From Müller, G.; Schwahn, D.; Eckerlebe, H.; et al. Chem. Phys. 1996, 104, 5326–5337.28

Structure Characterization in Fourier Space | Neutron Scattering dependent structure factor σðQ; tÞ ¼ SðQ; tÞ=ST0 ðQÞ for times between 2 and 238 h. The solid lines represent fits of eqns [54]– [56] and demonstrate their limitation to the early stages of spinodal decomposition which, however, extends at least over the first 70 h. Finally, after 238 h of phase separation the system is in the nonlinear regime as can be seen from a second-order peak at 4  10−2 nm−1. In order to perform the SANS experiments, the samples were quenched to room temperature, which is far below the glass transition temperature at about 100 °C and which allowed measurements with sufficiently good statistics. For a complete analysis of S(Q, t) in terms of eqn [54] it is essential to know the two-structure factors at T0 and at Tf. Above the critical temperature at T0 after sufficiently long relaxation time, S(Q) represents the equilibrium fluctuation modes for all Q. This is not the case for temperatures below TC. During the early stages of spinodal decomposition, long wavelength thermal fluctuation modes are unstable, for exam­ ple, fluctuation modes at Q values smaller than a critical QC drive the process of phase decomposition. On the other hand, short wavelength fluctuation modes, for example, Q > QC are stabilized by the surface energy expressed as a gradient energy term in the free enthalpy of mixing (see eqn [57] below).25 These stable fluctuation modes are described by the Zimm approximation as demonstrated by the two straight lines for T = 170 and 150 °C in Figure 21. At small Q below the so-called critical QC, the Zimm approximation becomes negative repre­ senting the unstable fluctuation modes. The structure facture below TC is also called ‘virtual’ structure factor.25 During the early stages of spinodal decomposition, the unstable fluctuation modes are small and noninteracting (linear approximation) and, consequently, the sample can approxi­ mately be regarded as in a homogeneously mixed state. Positive and negative susceptibilities in Figure 21 are consistent with Gibbs stability conditions as, according to the fluctuation-dissipation theorem in eqn [53], S(0) is equivalent to the second derivative of the Gibbs free enthalpy of mixing with respect to the composition. In Figure 22(a), the inverse susceptibilities of the 1.42  106 cm3 mol−1 blend and of a 2 times smaller dPS/PS system are plotted versus inverse temperature

T = 240 °C

S −1(Q) (10−6mol cm−3)

6

T = 170 °C T = 150 °C

4

2

0

0

2

4

Q 2 (10−3 nm−2) Figure 21 Time-dependent structure factor of the early stage of spinodal decomposition. From Müller, G.; Schwahn, D.; Eckerlebe, H.; et al. Chem. Phys. 1996, 104, 5326–5337.28

343

above and below TC. The critical temperatures of the larger and smaller systems are 231.3 and 130.3 °C, respectively. They were determined from the point of intersection according to S−1(0) = 0. The inverse susceptibilities from both samples follow a straight line in accordance with mean-field approximation (eqn [54]) and deliver the same FH parameter as shown in Figure 22(b). This result is an experimental consistency check for the determination of the equilibrium structure factor below the critical point, as the lower molecular dPS/PS sample was mostly measured in the one-phase region. It confirms the concept of a virtual structure factor that correctly determines the susceptibility according to eqn [52]. It furthermore shows that the FH parameter is a segmental quantity. From the knowledge of the equilibrium structure factors, the dynamical structure factor L(Q, t) is determined (eqn [49]). The diffusion constant and the Onsager coefficient is found from the relaxation rate (eqn [56]). Both parameters are dis­ played in Figure 23(a) and 23(b). The self-diffusion constant of the two blends are shown in an Arrhenius plot representa­ tion. The derived activation energy is the same for both samples and in agreement with other work.29 The diffusion constant follows the scaling behavior D0 ∝ V−2 as predicted for the repta­ tion model for polymers.22 In Figure 23(b), the range of interaction of the Onsager coefficient, as derived from eqn [56], is plotted versus time. At zero time, RΛ ≅ Rg as predicted; however, RΛ increases with time and depending on tempera­ ture to nearly twice of Rg. At higher temperatures, RΛ starts to increase earlier. This increase of RΛ leads to a nonexponential growth behavior of the dynamical structure factor L(Q, t) in contrast to the prediction of Cahn–Hilliard–Cook theory (eqn [54]). The observed increase of the Onsager interaction range is interpreted as a coupling of the composition fluctuations and stress fields in the sample during phase separation.30 A similar analysis of the early stage of spinodal decomposition has been performed for a dPS/poly(vinyl methylether) (PVME) sample.31

2.11.3.2.2 Deviation from Flory–Huggins theory: Critical phenomena FH theory and RPA are mean-field theories that properly describe the thermodynamics of polymer blends at sufficiently high temperatures above the critical point. In this range, ther­ mal fluctuations are weak such that they may be described within the Gaussian approximation.32 On the other hand, when approaching the critical point, thermal composition fluc­ tuations become strong, leading to nonlinear effects. Asymptotically, close to the critical temperature, they obey the critical universality class of 3D-Ising behavior. In eqn [57], 3D-Ising behavior is described by the Ginzburg–Landau Hamiltonian as a functional of the spatially varying order parameter ð n o H ¼ dd x c2 ½∇ΦðxÞ2 þ rΦ2 ðxÞ þ uΦ4 ðxÞ ½57 The order parameter Φ(x) represents the volume fraction Φ of one of the two polymer components. The first term in the right-hand side expression describes the surface energy and leads to the correlation length of thermal fluctuations, whereas the second term corresponds to the susceptibility according to S(0) = 1/2r, and the third term describes nonlinear effects from

344

Structure Characterization in Fourier Space | Neutron Scattering

(a)

(b) 3.5

4 dPS/PS ΦC = 0.5

dPS/PS Φ = 0.5

3.0

VW = 0.7 × 106 cm3 mol−1 VW = 1.42 × 106 cm3 mol−1

2 1

Γ (10–6 mol cm–3)

S–1(0) (10–6 mol cm–3)

3

TC = 231.3 °C

0 TC = 130.3 °C

−1 −2

2.5 2.0 1.5 1.0

VW = 0.72 106(cm3 mol–1) VW = 1.42 106(cm3 mol–1)

0.5

−3 1.9

2.0

2.1

2.2 2.3 2.4 1/T(10–3 K–1)

2.5

2.6

0.0 1.9

2.0

2.1

2.2 2.3 2.4 1/T(10–3 K–1)

2.5

2.6

Figure 22 Susceptibility of dPS/PS for two molecular volumes in one- and two-phase regime. S−1(0) = 0 gives the critical temperature. Γ = [(2.76  0.07) 103/T – (4.05  0.16)]10−6mol cm−3. From Müller, G.; Schwahn, D.; Eckerlebe, H.; et al. Chem. Phys. 1996, 104, 5326–5337.28

10–13

800

EA = (139±5) kJ mol–1

700

VW = 1.42 106 (cm3 mol–1)

10–14

RΛ (Å)

D0 (cm2 s–1)

VW = 0.72 106 (cm3 mol–1)

10–15 10–16 10–17 2.0

170 °C 190 °C 210 °C 220 °C

600 500

Rg

400

2.1

2.2 2.3 2.4 1/T (10–3 K–1)

2.5

2.6

0.1

1

10

100

Time (h)

Figure 23 (a) Self-diffusion constant in Arrhenius presentation of two dPS/PS blends. (b) Range of interaction of Onsager coefficient. From Müller, G.; Schwahn, D.; Eckerlebe, H.; et al. Chem. Phys. 1996, 104, 5326–5337.28

the fluctuation fields Φ(x).32 Within Gaussian approximation, the last term is negligible. Within the mean-field and 3D-Ising regimes the susceptibil­ ity S(0) is described by simple scaling laws of the reduced temperature τ = 1 − TC/T (TC critical temperature) according to Cτ−γ with the critical amplitude C and the critical exponent γ. The critical exponents γ are known to be equal to γ = 1 and 1.24 in mean-field and 3D-Ising regimes, respectively. The mean-field critical amplitude is expressed in terms of FH parameter and molar volume 1/CMF = 2Γh/TC = 2(Γσ + 2/V) (Figure 24). Mean-field and 3D-Ising behavior are observed in the limits where the system is asymptotically far and close to the critical temperature, respectively. Analytical expressions for crossover functions have been elaborated that describe susceptibility and correlation length over the full miscible range of the blend and thereby bridge both asymptotic limits. Such crossover func­ tions will not be discussed here as they have been reviewed by Sengers and co-workers33 and in context with polymer blends by Schwahn.34 The temperature of transition TX between mean-field and 3D-Ising regimes is estimated by the Ginzburg number Gi according to the criterion Gi ¼ 0:069ðCþ =CMF Þ1=ðγ−1Þ

½58

Gi is determined in terms of the ratio of the Ising and mean-field critical amplitudes of the susceptibility and the exponent 1/(γ−1). The exponent is about 4 as determined from the Ising critical exponent and Gi is identified as the reduced temperature of TX, according to 1 − TC/TX. The mean-field critical amplvitude of polymer blends was already expressed in this section, whereas the 3D-Ising ampli­ ð2−γÞ tude Cþ ∝Vchain was derived by Binder23 for symmetric blends with molar volume Vchain. Both expressions lead to the Ginzburg criterion for polymer blends ð2−γÞ

Gi ∝ ½Vchain ð2=Vchain þ Γσ Þ1=ðγ−1Þ

½59

In the case of a zero segmental entropy Γσ eqn [59] becomes a universal scaling law where Gi ∝ 1=Vchain . The last expression was originally derived by de Gennes assuming incompres­ sible polymer melts.22 From eqn [59], we observe that a positive entropic term Γσ in combination with the exponents 1/(γ – 1) ≅ 4.23 strongly enhances Gi and thereby the degree of thermal composition fluctuations near the critical temperature. Two examples of binary homopolymer blends exhibiting a crossover from mean-field to 3D-Ising critical behavior will be presented. The first blend is a lower critical solution tempera­ ture (LCST) blend of high-molar-volume dPS and PVME at

Structure Characterization in Fourier Space | Neutron Scattering

Critical point

Binodal

Spinodal Composition Φ Figure 24 Schematic phase diagram of a binary polymer blend. Thermal composition fluctuations are usually measured at critical composition along the dashed line. Mean-field approximation and 3D-Ising behavior is found far above and near the critical temperature, respectively. The posi­ tion of the crossover regime is estimated by the Ginzburg criterion.

critical composition. LCST blends are miscible at low tempera­ ture and become unstable with respect to miscibility at high temperature. The inverse susceptibility was plotted versus inverse temperature and analyzed with a crossover function as shown by the solid line (Figure 25(a)). From the parameters of the crossover function, the critical temperature, the Ginzburg number Gi = (4.5  0.5)10−5 and the mean-field and 3D-Ising critical amplitudes were derived from which the FH parameter was evaluated. The positive slope of S−1(0) is related to a negative enthalpic Γh (eqn [49]) that expresses the preferred energetic contact of the PS and PVME monomers. Phase separa­ tion in LCST systems is an entropically driven process and possible only when the amplitude of the negative entropy of mixing (Γσ < 0) is larger than the always positive combinatorial entropy of mixing. A second example is visualized in Figure 25(b) of two conventional upper critical solution temperature (UCST)

(a) 3.0 S –1(0) (10–5 mol cm–3)

2.11.3.3.1

2.0

Crossover

1.5 0.6

Ising

0.4

2.33 1/T

2.320

2.34 (10–3

2.5

K–1)

τ1.24

2.0 1.5 1.0

τ1

0.5

Meanfield 0.0

dPB(1,4)/PS dPB(1,4; 1,2)/PS

3.0

Crossover

0.2

2.32

4.0 3.5

Meanfield

1.0

Isotropic Lifshitz system

Mixing partially miscible (A/B) homopolymers with small amounts of a (A-B) diblock copolymer leads to an improved miscibility in a similar way as amphiphilic molecules do in oil–water mixtures. At larger diblock content, microemulsion

Ising

2.5

0.5

−1

2.11.3.3 Phase Behavior of (A/B) Polymer Blend/(A-B)

Diblock Copolymer Mixtures

(b) dPS/PVME

0.0

dPB/PS blends of critical concentration with 7% and 54% vinyl contents for the PB (polybutadiene) component and about 2000 cm3 mol−1 average molar volume. The experimen­ tal data were also fitted by a crossover function (solid line), whereas the dashed lines represent the asymptotic scaling laws corresponding to mean-field and 3D-Ising behavior. Both blends show a larger crossover range of the order of 13 K in comparison with the 1-K range for the high molecular dPS/PVME blend. The phenomenon of crossover behavior, for example, deviation from FH mean-field behavior was first observed for the large-molar-volume dPS/PVME blend.35 The observation of a crossover to the Ising case was surprising at that time because on the basis of the ‘incompressible’ Ginzburg criterion a temperature range of roughly 0.05 K was expected. Figure 26 shows a collection of Ginzburg numbers from various polymer blends versus their mean degree of polymer­ ization 〈N〉.36 A variation of Gi over four orders of magnitude is shown for samples ranging over 2 orders of magnitude in 〈N〉. Gi is distributed within an area confined by two N−1 and N−2 scaling laws, the first one proposed by de Gennes22 and the second one determined from SANS experiments. A remarkable observation is related to the order of magnitude larger Gi that is found for blends composed of shorter chains between N = 10 and 100 as compared to the Gi ≅ 10−2 of low molecular liquids. This observation in context with eqn [59] clearly demonstrates that the entropic term Γσ and thereby the compressibility plays an important role for polymer mixtures and is responsible for the much enhanced nonuniversal character of the variation of Gi with N. These observations, namely the limited applicability of FH theory, must be also considered for ordinary partially miscible UCST polymer blends whose chains are usually not larger than of the order of N = 100.

S –1(0) (10–4mol cm–3)

Temperature

Gi: = (1–TC/TX)

2.325

2.35

2.36

345

0.0

2.60 2.65 2.70 2.75 2.80 2.85 2.90 1/T (10–3 K–1)

Figure 25 (a) S (0) plotted versus 1/T. The solid lines describe the 1critical behavior of the thermal composition fluctuations. Near the critical temperature one observes deviations from the mean-field behavior. The spinodal and critical temperatures are determined from the extrapolated S−1(0) = 0. From Schwahn, D.; Janssen, S.; Springer, T.; et al. J. Chem. Phys. 1992, 97, 8775–8789.31

346

Structure Characterization in Fourier Space | Neutron Scattering

10

100 dPB(1,4)/PS dPB(1,4)/PS(150MPa) dPB(1,4;1,2)/PS dPB(1,4;1,2)/PS(200MPa) dPB(1,2)/PS dPB(1,2)/PS(150MPa) PPMS/d-PS PPMS/d-PS(100MPa) PDMS/PEMS d-PS/PVME

10–2

8 S(Q)(103 cm3 mol–1)

Ginzburg number

10–1

N –2

10–3

Diblock content: 3% (97.1 °C) 7.5% (97.1 °C) 20% (95.1 °C)

N –1

10–4

6

4

2 10–5

101

102 103 Degree of polymerization (N)

104 0

Figure 26 Ginzburg number of various polymer blend versus their

degree of polymerization N. The open symbols show the reduction of Gi in external pressure field. From Schwahn, D.; Meier, G.; Mortensen, K.; et al.

J. Phys. II (France) 1994, 4, 837–848.36

130

0

1

2

3

4

5

6

7

8

–2Å–1)

Q (10

Figure 28 Structure function from three characteristic-disordered regions of a Lifshitz system. From Pipich, V.; Schwahn, D.; Willner, L. Phys. Rev. Lett. 2005, 94, 117801; J. Chem. Phys. 2005, 123, 124904– 124916; Pipich, V.; Willner, L.; Schwahn, D. J. Phys. Chem. B 2008, 112, 16170–16181 (Part of the ‘Karl Freed Festschrift’).40

Double critical point LL transition point

120

Figure 27 Temperature – diblock concentration plane of the {PB/PS} blend phase diagram. The composition of the PB/PS homopolymer blend was the critical one of the binary blend. The right figure shows the ‘Lifshitz’ part. Meaning of symbols: (,) line of critical points with (−) the double critical point; (ξ) and (ψ) Lifshitz line between disordered and microemulsion phases (DμE and BμE droplet and bicontinuous microemulsion), respectively; (B) transition from disordered to micro emulsion phase, (χ) ‘Lifshitz transition point’ LLT, (Λ) Ordering transition to lamellar phase. From Pipich, V.; Schwahn, D.; Willner, L. Phys. Rev. Lett. 2005, 94, 117801; J. Chem. Phys. 2005, 123, 124904–124916; Pipich, V.; Willner, L.; Schwahn, D. J. Phys. Chem. B 2008, 112, 16170– 16181 (Part of the ‘Karl Freed Festschrift’).40

diblock copolymer of about 6 times larger molar volume pffiffiffiffiffiffiffiffiffiffiffi according to the parameter α ¼ VA VB =VA−B ≅0:16. At high temperature, one has two homogeneous (disordered) phases, which are separated by the Lifshitz line (LL). At concentrations below the LL (Φ < ΦLL), the structure factor S(Q) looks similar to that of binary blends with its maximum at Q = 0. Above the LL (Φ > ΦLL), one observes the characteristics of diblock copo­ lymer melts with the maximum of S(Q*) at the finite Q = Q*. Scattering patterns from three characteristic disordered regions of the phase diagram are shwon in Figure 28. At 3% and 20% diblock contents, one has the characteristic scattering patterns of homopolymer blends and diblock copolymer melts, respec­ tively, whereas at 7.5% one is near the LL, the border line of transition from blend to diblock like behavior. In all cases. S(Q*) represents a susceptibility with Q* = 0 and Q* ≠ 0 below and above ΦLL, respectively. The scattering contrast of these samples was chosen in a way that one of the two A or B monomers of the homopolymer and of the copolymer was deuterated with the same relative amount of deuterium whereas the other one protonated. Under such conditions, the structure factor S(Q) measures thermal compo­ sition fluctuations with respect to the total monomer fraction and which corresponds to a scalar order parameter represented by the local concentration Φ = Φ(x) of the A or B monomers. The basic thermodynamic features of those systems are described by the Ginzburg–Landau Hamiltonian according to ð n o H ¼ dd x c2 ½∇ΦðxÞ2 þ c4 ½∇2 ΦðxÞ2 þ rΦ2 ðxÞ þ uΦ4 ðxÞ ½60

and lamellar ordered phases are formed as visualized in the temperature–copolymer concentration plane of the (dPB/PS) polymer blend phase diagram of Figure 27. The dPB/PS blend of critical composition was mixed with a symmetric dPB/PS

representing an extension of the 3D-Ising functional (eqn [57]) that includes a higher expansion parameter of the gradient energy term.32,37 A principal effect of the dissolved diblock copolymers is the reduction of the surface energy, which

110

Disordered ‘Homopolymer’

Disordered ‘Diblock copolymer’

Temperature (°C)

100

LL

90 80 70 60 50

Two phase

DμE BμE Lamella

40 30

5

10 3 Diblock content (10–2)

40

Structure Characterization in Fourier Space | Neutron Scattering

Deviations from mean-field behavior were observed for the lower-molar-weight (PB/PS) and (poly(ethyl ethylene) (PEE)/ poly(dimethylsiloxane) (PDMS)) blends.40 The (PB/PS) mix­ ture (Figure 27) exhibits the characteristics of diblock copolymers between 6% and 8% diblock content at low and high temperatures, that is, S(Q) shows a maximum at a finite Q*, whereas at intermediate temperatures the behavior of homopolymer blends, that is, S(Q) has a maximum at Q = 0 is found. Apparently, the LL was crossed twice as its concentra­ tion changes with temperature near the two-phase region. The apparent nonmonotonic shape of the LL was explained by thermal composition fluctuations. Renormalization group cal­ culations by Kudlay and Stepanow44 showed that at high and low temperatures, ΦLL approaches the constant mean-field value when thermal fluctuations become negligible, and ΦLL changes with temperature when fluctuations become strong. A double critical point (DCP) as observed in Figure 27 represents an endpoint of a UCST and LCST line of critical points. The LCST critical line is observed over a limited diblock concentration of about 1%. Thus, the homogeneous phases above and below the two-phase regime must represent differ­ ent phases, as the compatibility is achieved by the entropic and enthalpic term of the Gibbs free energy of mixing, respectively; the SANS experiments show that the homogeneous phase at lower temperature is represented by a droplet microemulsion phase which must represent a more ordered state than the other two phases at higher temperature. Double critical points show a strong increase of γ if one applies the scaling law of S−1 (0) = C−1τγ with the ‘conventional’ reduced temperature field. If one, however, applies a modified reduced temperature field according to τ ¼ ð1−TUC =TÞð1−TLC =TÞ; then the inverse sus­ ceptibility delivers a critical exponent γ of the corresponding universality class, which here is of the isotropic Lifshitz class.45 At the DCP, one has TC = TUC = TLC and therefore S−1(0) = C−1τ2γ (Figure 29).

1.8

γ = 1.62

1.6

γ

according to the Hamiltonian is described by a smaller para­ meter c2. This parameter is positive at low copolymer concentration, becomes zero at the LL, and is negative for larger copolymer contents. The multicritical Lifshitz critical point occurs for r = 0 and c2 = 0. The coefficients u and c4 must be positive for stability reasons. The LL is estimated at a constant concentration accord­ ing to ΦLL = 2α2/(1 + 2α2) with the Lifshitz point (LP) at the lower end, representing a point of coexistence of the two dis­ ordered and two ordered phases. Three parameters characterize the LP, namely (n, d, m), which have the meaning of number of components of the order parameter, the dimension of space and wave vector Q*, respectively. The order parameter of poly­ mer blends is isotropic (d = m = 3) and scalar (n = 1). The LL of the present system should be observed at ΦLL = 0.048. The experimental LL, however, shows systematic deviations and no LP is found as predicted from mean-field approximation; the LL is found at slightly larger diblock concentration and shows a pronounced dependence on temperature near the two-phase boundary. The phase diagram near LL is determined by a breakdown of mean-field approximation as thermal com­ position fluctuations become strong. Another argument for this rather complex phase behavior is that the two constituents belong to different universality classes. Diblock copolymers are of Brasovskii type;38 they decompose into spatially modu­ lated phases of mesoscopic length scale determined by the polymer size, while homopolymer blends are of 3D-Ising type and decompose into macroscopically large domains. A first systematic study of such system was performed on the relatively large-molar-mass symmetric polyolefins PE and PEP and the corresponding diblock copolymer PE–PEP; PE being polyethylene and PEP being poly(ethylene propylene).39 A mean-field Lifshitz like behavior was observed near the predicted isotropic Lifshitz critical point with the critical expo­ nents γ = 1 and ν = 0.25 of the susceptibility and correlation length, and the structure factor following the characteristic mean-field Lifshitz behavior according to S(Q) ∝ Q−4. Thermal composition fluctuations were apparently not so rele­ vant as indicated by the observation of mean-field critical exponents. On the other hand, no Lifshitz critical point was observed and instead a one-phase channel of a polymeric bicontinuous microemulsion phase appeared. Equivalent one-phase ‘channels’ were also observed in other systems.40,41 This class of blends has two relevant aspects. The first one has to do with the compatibility of two polymers within the channel of microemulsion phases and therefore may be of interest for practical application. The second aspect is the the­ oretical interest in Lifshitz systems with respect to critical universality, as reviewed in Reference 42. Near the Lifshitz critical point, thermal composition fluctuations are expected to become strong because of reduced surface energy (c2 in eqn [60]), leading to a lower threshold force for thermal fluctua­ tions. The difficulties in the evaluation of critical exponents can be understood from the large upper critical dimension dU above which the mean-field approximation becomes valid. In particular, one gets dU = 8 for isotropic LP (d = m) according to the relationship dU = 4 + m/2, which is appreciably larger in comparison with dU = 4 of 3D-Ising systems and the spatial dimension d = 3 of the system under investigation. Critical exponents of isotropic LP so far could only be determined for systems with spatial dimension near d = 8.43

347

1.4

γIsing = 1.24 1.2 Φ DCT 1.0

0

1

2 3 4 5 6 Diblock content (10–2)

7

8

Figure 29 Crossover from 3D Ising to isotropic Lifshitz class of critical universality. From Pipich, V.; Schwahn, D.; Willner, L. Phys. Rev. Lett. 2005, 94, 117801; J. Chem. Phys. 2005, 123, 124904–124916; Pipich, V.; Willner, L.; Schwahn, D. J. Phys. Chem. B 2008, 112, 16170– 16181 (Part of the ‘Karl Freed Festschrift’).40

348

Structure Characterization in Fourier Space | Neutron Scattering

In the blend-like regime, a crossover from 3D-Ising to iso­ tropic critical Lifshitz behavior was observed at about 4.8% diblock concentration as indicated by the dashed area in the phase diagram and the susceptibility critical exponent γ in Figure 29. A crossover of the exponent γ from the Ising 1.24 to larger value of γ = 1.62 is visible in coincidence with results from earlier SANS studies on PEE/PDMS.40 Up to now, the Lifshitz critical exponents could only be determined from the SANS experiments. A theoretical determination was not possi­ ble yet because the theory lacks a small expansion parameter caused by the large upper critical dimension (dU = 8).43

2.11.3.4

Quenched SANS

SANS may be also be used in order to investigate time-dependent process in taking snapshots of the polymer structures starting from a nonequilibrium state. This may either be done by in real time, for example, by stopped flow experi­ ments,46 or in quenching experiments where relaxation steps are frozen in after controlled relaxational periods. As a further example, we want to address molecular processes behind the nonlinear rheology of long-chain polymers and present results on the initial chain relaxation after an applied step strain. A large step extensional elongation is expected to affinely deform the chain contour. This will cause the chain radius of gyration parallel to the flow to increase and conversely the one in the perpendicular direction to decrease. After cessation of the strain, the first relaxation processes is expected to be a retraction of the chain within the still affinely deformed tube. The longest time scale of this process should be the equilibration time τR (Rouse time) of the chain along the tube. By a fluctuation-dissipation theorem, this mechanism is related to

(a)

the chain contour length fluctuations (CLFs). Chain retraction in the elongated tube should reduce the radius of gyration in all is expected to attain a minimum, directions. Therefore, Rperp g manifested as an initial decrease followed by a subsequent increase in the perpendicular scattered intensity at some time after the deformation, determined by the Rouse time, before diffusive mechanisms return it to equilibrium. High-molecular-weight (250 kg mol−1) polyisoprene mix­ tures of hydrogenated and deuterated molecules were uniaxially stretched and rapidly quenched. The SANS pattern of the initial deformed state served as a reference to assess the subsequently annealed samples at relaxation times of 0.4, 3.2, 8.4, and 15.7 times the Rouse time τR. To perform the experi­ ments, the melt was quenched, studied, and thereafter relaxed stepwise in a controlled way. After each relaxation, the sample was quenched again and studied by SANS.47 The plots of scattered intensity against scattering vector (Figure 30) demonstrate that, throughout the relaxation regime, the anisotropy is strong at large length scales and weakening as the scale of the tube diameter is reached at higher Q. As relaxation progresses, features at higher Q relax to iso­ tropy faster than at lower Q. At t = 3.22τR, the parallel scattering shows significant relaxation relative to the fully affine curves (dotted lines). However, throughout this time interval, the perpendicular scattering remains close to the affine curve over almost the entire range of scattering vectors (Figure 30(a)–30(c)). In particular, a small but significant increase of the scattered intensity between t = 0 and 0.4τR is visible in the perpendicular component of the scattering due to a shift to higher Q. It indicates the presence of a shrinking Rg in the per­ pendicular direction. The data were quantitatively evaluated in terms of stochastic microscopic evolution equation for the

(b)

100

100

10–1

10–2

10–3 (c)

t = 0.4τR Intensity/I0

Intensity/I0

t=0

Parallel Perpendicullar Affine Theory

10–1

q ((nm)–1)

10–2

10–3

100

100

10–1

(d)

10–1

q ((nm)–1)

100

10–2

10–3

t = 15.7τR Intensity/I0

Intensity/I0

t = 3.2τR 10–1

10–1

q ((nm)–1)

100

100

10–1

10–2

10–3

10–1

q ((nm)–1)

100

Figure 30 Comparison of measured and predicted normalized structure factors after 0, 0.4, 3.2, and 15.7τR (a–d, respectively). The solid lines are predictions of the theory of Graham et al.,48 and the dotted lines correspond to a completely affine deformation on all length scales. From Blanchard, A.; Graham, R. S.; Heinrich, M.; et al. Phys. Rev. Lett. 2005, 95, 166001.47

Structure Characterization in Fourier Space | Neutron Scattering

SANS dat perpendicular

R gperp/Rg0

0.8

0.78

0.76

0.74 0.001

0.01 0.1 1 10 Time since beginning of deformation/τR

100

Figure 31 Data and theory for relaxation of the radius of gyration perpendicular to the stretch direction after deformation and experimental perp data highlighting the nonmonotonic region of Rg . From Blanchard, A.; Graham, R. S.; Heinrich, M.; et al. Phys. Rev. Lett. 2005, 95, 166001.47

dynamics of the space curve describing the tube contour.48 Similar to the Warner–Edwards approach49 for polymer net­ works, the chain fluctuations on the level of tube were induced. The solid lines in Figure 30 present the results of this modeling. A plot of the deformed radius of gyration with time (Figure 31) portrays the SANS signature of chain retraction in ðtÞ, providing a terms of a pronounced minimum in Rperp g microscopic confirmation of the tube retraction concept. The (a)

349

theory demonstrates that the time of occurrence and depth of this minimum are consistent with the retraction process, although the model systematically underpredicts the degree of deformation by 4% of the total deformation. Thus, using well-controlled quenched SANS experiments, the chain relaxation in the nonlinear regime can be elucidated over a broad range of length scales including the experimental manifestation of the long-predicted chain retraction process. Modeling these data provides a significant quantitative test for molecular theory and illustrates the possibility of linking mole­ cular structure and processing history. Along the same line, the complex hierarchical relaxation of branched polymers may be scrutinized on a molecular scale. The rheological properties of polymer melts are strongly modified by chain branching. Branched polymers exhibit a significantly broader spectrum of relaxation times, compared with their linear counterparts. Furthermore, their nonlinear rheology is altered significantly. Given its great importance for polymer processing, a quantitative molecular understand­ ing is one of the great challenges of polymer science. So far, attempts to create a molecular rheology for branched polymers on the basis of the tube concept have been partially success­ ful.50 We briefly describe SANS experiments on the controlled relaxation of H-polymers (polymers with an H shape), where for the first time, relaxation experiments on branched polymers on the molecular scale were attempted.51 Figure 32 displays the morphology of an H-polymer, including the confining tube for two different stages of relaxation. In the framework of the tube

(b)

(c) 10

d Σ (q ) (cm−1) dΩ

8 6 4 2 0 0 0.02 0.04 0.06 0.08 0.10 0.12 q (Å−1)

d Σ (q ) (cm−1) dΩ

10 8 6 4 2 0 0 0.02 0.04 0.06 0.08 0.10 0.12 q (Å−1) Figure 32 (a) Small-angle neutron scattering intensities for a stretched H-polymer sample (2D detector image). (b) Intensity cuts along the directions parallel to the deformation (open circles) and perpendicular to the deformation (open squares) compared with theoretical calculations51 for the sample strained to λ = 2 after annealing times ‘0’ (top graph) and 6  102 s (bottom graph) at 25 °C (the sample was quenched to –85 °C). The term d∑/dΩ(Q) represents the absolute macroscopic cross section. (c) H-polymer in its confining tube immediately after deformation (top schematic), with the arms fully confined, and after 6  102 s (bottom schematic), when the arms have relaxed by 12%. From Heinrich, M.; Pyckhout-Hintzen, W.; Allgaier, J.; et al. Macromolecules 2002, 35, 6650–6664.51

Structure Characterization in Fourier Space | Neutron Scattering

model, the relaxation of an H-polymer has to take place in a hierarchical fashion. Since the crossbar can move only if the arms have retracted from their confining tubes, CLFs play a key role in the relaxation. A number of relaxation stages were proposed. 1. At early times, CLFs of the arms take place in a similar way as for linear chains. 2. For deeper arm fluctuations, in which larger sections of the arm contract closer to the branching point, an increasingly important entropic barrier has to be crossed, slowing down the process exponentially; at the same time, the liberated arm segments dilute the tube confine­ ment of the crossbar. 3. After complete arm retraction, the crossbar dynamics take over, starting with CLFs of the branch points, where all of the friction is concentrated. 4. The crossbar dynamics continues with reptation within the diluted tube. In these terms, a quantitative theory for G(ω) was worked out.50 5. Recently, a study on a melt of H-polymers, where the arm tips were labeled using deuterium, has been performed. The polymer was stretched by different amounts (twofold, three­ fold: λ = 2, λ = 3, where λ is the chain extension factor). Thereafter, as for the case of the arm retraction, the melt was studied after controlled annealing and requenching in several steps. In parallel, the dynamic modulus was studied. Both G′′(ω) as well as the SANS data were attempted to be described in the same theoretical frame. At the reference temperature of T = 20 °C from rheology, stages (1) and (2) were expected within a time frame of 10−5–102 s, while the crossbar motion was predicted to take place between 101 and 105 s. Figure 32 presents SANS results for λ = 2. The studied relaxation times corresponded to stages (1) and (2) in the early stages of polymer relaxation. The lines display a fit with the predictions of the tube theory. Although the data showed a good fit with the theory, a number of important discrepancies between the scattering data and the rheological reference became manifest: (1) rheology overestimated the amount of arm relaxation by a factor of 3 or 4; (2) the undiluted tube diameter of d = 3.3 nm is much smaller than the rheological value of d = 5.1 nm; and (3) the best fits were obtained by disregarding any arm retraction or branch point withdrawal within the stretched tube – both should occur in nonlinear response. The discrepancies between theoretical prediction and experiment at later stages (not shown here) are even more important. In particular, a growing anisotropy with increasing relaxation time was observed that theoreti­ cally cannot be accounted for. These first experimental results demonstrate the ability of neutron scattering together with careful synthesis and new theoretical developments to take up the challenge and to scru­ tinize the molecular process underlying the complex rheology of branched polymers. Further experiments and theoretical progress in dealing with the very difficult treatment of a com­ bination of annealed and quenched variables will eventually lead to a true molecular rheology.

2.11.4 Polymer Dynamics 2.11.4.1

The Rouse Model

The standard model of polymer dynamics, the Rouse model,52 treats the dynamics of a Gaussian chain in a heat bath. Thereby it implicitly assumes that on the lengths and timescales considered all forces from local potentials, which are related to the indivi­ dual chemical nature of a given polymer have already decayed. Then only entropic forces originating from the conformational chain entropy in combination with the forces generated by the heat bat (i.e., friction and the associated stochastic kicks) drive the dynamics. With these scales QRE > 1, the normalized single-chain dynamic structure factor S(Q, t)chain for this model pffiffiffiffiffiffiffiffiffiffiffi can be written as a function of a scaling variable u ¼ Q2 Wℓ 4 t combining spatial and temporal scales53 8 ð∞ ð∞ < u cos ð6xs=uÞ ds exp − s − Schain ðuÞ ¼ : 3π x2 0 0 9 = � �  1 − expð−x2 Þ dx ½61 ; Thereby W = 3kBT/(ζℓ2) is the elementary Rouse frequency. It is given by the ratio of the entropic force 3kBT/ℓ2 and the friction coefficient ζ.ℓ2 is the mean squared segment length. Equation [61] is only valid in the asymptotic Q−2 tail of the scattering function. A full treatment54 yields the dynamic struc­ ture factor also in the low Q limit. At t = 0, this leads to the well-known Debye function (eqn [20]). Figure 33 displays the temporal evolution of the ‘quasielastic SANS’. The uppermost curve corresponds to the Debye function, while the lower curves visualize the decay of the structure factor due to the Rouse relaxation. For t → ∞, all internal chain correlations are lost and the structure factor displays the Gaussian density profile within a polymer coil. We note that the polymer coil in addition to the Rouse modes is also subject of translational diffusion. In Gaussian approximation (eqn [3]), the Sself(Q, t) directly reveals the mean square displacement (MSD). In the Rouse model,55 �

� � � 3k B T 1=2 R2 ðtÞ ¼ 6Dcm t þ 2ℓ2 2 t πζ ℓ

½62

1.0 0.8 S(Q , t)

350

0.6 t=0

0.4 t=∞

0.2

0

0

2

4 Q Rg

6

8

Figure 33 Development of S(Q, t) for Rouse motion for different times disregarding chain translational diffusion.

Structure Characterization in Fourier Space | Neutron Scattering

the observed spectra from PEE (90% dPEE, 10% hPEE) with a molecular weight of ðMhw ¼ 21:5 kg ml−1 ; Mdw ¼ 24:5 kg mol−1 Þ and a narrow-molecular-weight distribution.57 The solid lines give the prediction of the dynamic structure factor of eqn [61]. Obviously very good agreement is achieved.

Q = 1.0 nm−1

�R 2 (t ) nm−2�

101

2.11.4.2

100 t 1/2 10−1

100 Fourier time t / ns

101

Figure 34 Data of h-PEP in the representation of 6ℓn[Sinc(Q, t )]/Q2 vs. time for T = 492 K. Solid lines describe the asymptotic power laws. From Wischnewski, A.; Monkenbusch, M.; Willner, L.; et al. Phys. Rev. Lett. 2003, 90, 058302.56

where Dcm ¼ kB T=Nζ is the center of mass diffusion coefficient. Figure 34 displays the time-dependent MSD 〈R2(t)〉 obtained from a high-molecular-weight (Mw = 80 000) monodisperse PE–PEP melt at 492 K.56 Following eqn [3], the MSD was calculated as �

� 6ℓn Sself ðQ; tÞ R2 ðtÞ ¼ − Q2

½63

it follows with high accuracy the predicted square root law in time (for the high Mw polymer, the translational diffusion does not play any role). Since neutron quasielastic scattering resolves dynamic processes in space and time, these measure­ ments give direct information on the segment displacement at a given time, for example, at 10 ns 〈R2(10 ns)〉 = 620 Å2 (i.e., the average proton has traveled 25 Å during this time interval). The pair correlation function arising from the segment motion within one given chain is observed if some protonated chains are dissolved in a deuterated matrix. Figure 35 displays

PEE homopolymer, T = 473 K 1.0

0.05 0.08

Schain(Q, t)/Schain(Q)

0.8

0.10 0.121

0.6

0.4 0.187 0.2

Q/A−1

Rouse 0 0

351

20

10 t /ns

Figure 35 Single-chain structure factor from a PEE melt at 473 K. The numbers along the curves represent the experimental Q-values in Å−1. The solid lines are a joint fit with the Rouse model (eqn [61]). From Richter, D.; Monkenbusch, M.; Arbe, A.; et al. Adv. Polym. Sci. 2005, 174, 1–221.55

Reptation

For long chains, topological chain–chain interactions in terms of entanglements become important, and are dominating the dynamical behavior. In the reptation model, these constraints are described by a virtual tube, which localizes a given chain and limits its motion to a 1D Rouse motion inside the tube (local reptation) and a slow diffusive creep motion out of the tube (reptation).58 Applying NSE technique, it has become possible to observe the dynamic structure factor S(Q, t) asso­ ciated with tube confinement and local reptation.6 De Gennes59 and Doi and Edwards54 have formulated a tractable analytic expression for the dynamic structure factor. Thereby they neglected the initial Rouse regime, that is, the derived expression is valid for t > τe once confinement effects become important (τe is the entanglement time, the Rouse relaxation time of an entanglement strand). The dynamic structure factor is composed from two contributions Sloc(Q, t) and Sesc(Q, t) reflecting local reptation and escape processes (creep motion) from the tube, respectively: � � 2 2 �� Q d Schain ðQ; tÞ Sloc ðQ; tÞ ¼ 1− exp − 36 Schain ðQÞ � 2 2� Q d þ exp − ½64 Sesc ðQ; tÞ 36 "� � # � � t t 1=2 with d the tube diameter, Sloc ðQ; tÞ ¼ exp − erfc ; τ0 τ0 and τ 0 ¼ Wℓ364 Q4 . For short times Schain(Q, t) decays mainly due to local reptation (first term of eqn [64]), while for longer times (and low Q) the second term resulting from the creep motion dominates. The ratio of the two relevant time scales τ0 and τd is proportional to N3. Therefore, for long chains at intermediate times τe < t < τd, a pronounced plateau in Spair(Q, t) is predicted. Such a plateau is a signature for confined motion and is present also in other models for confined chain motion. In Figure 36, dynamic structure factor data from a Mw = 36kg mol−1 PE melt are displayed showing very clearly the tendency to form plateaus at high times.6 In the spirit of eqn [64] and neglecting the ongoing decay of Schain(Q, t) due to local reptation, from the heights of the achieved plateaus we may obtain a first estimate for the amount of confinement. Identifying the plateau levels with a Debye–Waller factor describing the confinement, we get a tube diameter of 45 Å, a value which is a lower estimate for the tube diameter, since Sloc (Q, t) is not fully relaxed. The dashed lines in Figure 36 are the predictions from this Debye–Waller factor estimate. The solid lines are a fit with eqn [64] that are in quantitative agreement with the experiment. The confinement effect also expresses itself in terms of the MSD. If it reaches the order of the tube diameter, then motional restrictions are expected. For the crossover time τe, we may take the relaxation time of a polymer section, spanning the tube diameter

352

Structure Characterization in Fourier Space | Neutron Scattering

1.00

101

0.75 0.50

0.46 0.28

0.25 0.00

Q = 1.0 nm−1 Q = 1.5 nm−1

0.71

0

50

100 t / ns

100

t 1/2 10−2

1 d4 ζ 0 1 d4 : 2 ¼ 2 2 π Wℓ4 3π kB T ℓ

For times t > τe, 1D curvilinear Rouse motion along the tube has to be considered leading to two different regimes of motion. For t < τR, the Rouse modes are still active. Therefore, we expect curve linear Rouse dynamics with 〈R2(t)〉curve  t1/2. At longer times, Rouse diffusion along the tube takes place with 〈R2(t)〉curve  DRt. In order to transform into laboratory space, we need to consider that due to the Gaussian conformation of the tube a mean square displacement along the tube gives rise to only the square root of this displacement in lab space. Thus, for t < τR, we have 〈R2(t)〉  t1/4, while for longer times 〈R2(t)〉curve  t1/2.60 In Gaussian approximation, the self-correlation function of a reptating chain would directly relate to the above-calculated mean square displacements. However, diffusion along the 1D tube contour is not a Gaussian process in the laboratory frame – the corresponding self-correlation function becomes non-Gaussian for t > τe. " rffiffiffiffiffiffiffiffiffiffiffiffiffiffi # � 4 2 2 � Q2 d 〈R2 ðtÞ〉 Q d 〈R ðtÞ〉 Sself ðQ; t > τ e Þ ¼ exp  erfc pffiffiffi 72 3 3 6 2 ½65 Where

erfc is the complementary error function, ð pffiffiffi ∞ −t2 erfcðxÞ ¼ 2= π e dt. We note that eqn [65] is strictly x

t 1/4

150

Figure 36 The dynamic structure factor from a Mw = 36 kg mol−1 PE-melt at 509 K as a function of time. The solid lines are a fit with the reptation model (eqn [64]). The Q-values are from above Q = 0.5, 0.77, 1.15, 1.45 nm−1. The horizontal dashed lines display the prediction of the Debye–Waller factor estimate for the confinement size. From Schleger, P.; Farago, B.; Lartigue, C.; et al. Phys. Rev. Lett. 1998, 81, 124–127.6

τe ¼

�R 2(t )� nm−2

S(Q, t )/S(Q )

0.86

valid only for t ≫ τe when 〈R2(t) ≫ d2〉. The effect on the scatter­ ing function is that if (wrongly) interpreted in terms of the Gaussian approximation the cross over to local reptation appears to occur at significantly lower values of τe. The general asymptotic t1/4 law remains untouched. The expected cross over from a t1/2 to a t1/4 law has been recently observed by NSE spectroscopy.56 Experiments were performed on PE samples at 509 K. The observed spectra were converted via eqn [63] toward mean square displacement. Thereby the Gaussian assumption is implied (Figure 37). Substituting the known Rouse rate Wℓ4 4 4 −1 (509 K) = (7  0.7)  10 Å ns in eqn [62], the solid line ~ t1/2 is obtained. It quantitatively corroborates the correctness of the Rouse description at short times. The data also reveal clearly a transition to a t1/4 law, though a straightforward

10−1

100 Time t/ns

101

Figure 37 Incoherent scattering data from PE in a representation of -6ℓn [Sself(Q, t)]Q2 which is the mean square displacement 〈R2(t)〉 as long as the Gaussian approximation holds. Solid lines describe the asymptotic power laws 〈R2(t)〉 ∝ t1/2; t1/4. Dotted lines: prediction form the Gaussian approximation, dashed lines: see text. From Wischnewski, A.; Monkenbusch, M.; Willner, L.; et al. Phys. Rev. Lett. 2003, 90, 058302.56

calculation would predict the dotted line. The discrepancy explains itself in considering the non-Gaussian character of the curve-linear Rouse motion. Fixing Wℓ4 and d to the values obtained from single-chain structure measurement (Figure 36), the dashed lines in Figure 37 reveal the prediction of the non-Gaussian treatment.

2.11.4.3

Reptation Limiting Processes

It is well known that a number of salient properties of entangled polymer melts are only qualitatively in agreement with reptation (see, e.g., Reference 50). These properties include (1) the viscosity η, which, in general, follows a power law η  M3.4 instead of η  M3.0 as predicted by reptation; (2) the translational diffusion coefficient, where D  M−2.3 instead of D  M−2 required by reptation is found; and (3) the frequency dependence of the dynamic loss modulus G″(ω), where at intermediate frequencies G′′(ω)  ω−1/4 is observed rather than G′′(ω)  ω−1/2 as expected. In order to cure the shortcomings, a number of additional relaxation processes have been invoked. The most prominent among them being CLFs and constraint release (CR). While CLF is an effect of the confined chain itself, CR stems from the movement of the chains building the tube, which of course undergo the same dynamical processes as the confined chain (see Figure 38). In the following sections, we first address NSE spectroscopy results on CLFs phenomena; thereafter we discuss recent results on CR.

2.11.4.3.1

Contour length fluctuations

The CLF effect evolves from the participation of the chain ends in the local reptation process. Any chain retraction and subse­ quent expansion will lead to a loss of memory of the original confinement of the tube. CLFs are a key mechanism for the relaxation at earlier times and also the basis for hierarchical relaxation processes of branched polymers, where CLFs are considered to be the fun­ damental process facilitating the release of side branches.

Structure Characterization in Fourier Space | Neutron Scattering

353

1.00

S(Q, t )/S(Q )

0.75

0.50

0.25

0.00 Figure 38 Schematic presentation of the CLF and CR mechanisms: chain end fluctuations lead to a shortening of the effective tube length, while the dissolving of entanglements allow chain motions beyond the initial tube constraints. From Zamponi, M.; Wischnewski, A.; Monkenbusch, M.; et al. Phys. Rev. Lett. 2006, 96, 238302.63

Mathematically, the problem is treated as a first passage pro­ blem. Whenever a tube contour s is visited by the free end, it ceases to exist. The functional form of the tube survival prob­ ability μ(t) has been derived from scaling arguments.50,61 Cμ Z

� � �1=4 t τe

2.11.4.3.2 ½66

where the numerical constant Cμ = 1.5  0.02 is obtained from stochastic simulations, Z = N/Ne is the number of entangle­ ments and Ne the number of segments forming an entanglement strand. Equation [66] provides quantitative knowledge on the chain fraction, which at a time t is still confined. All parameters are known from the NSE experiments on the dynamics of asymptotically long chains, where the CLF effect does not play a role. With this knowledge, an experiment was designed where the dynamic structure factor of a chain, which is subject to CLF, was compared with that of an identical chain, where the con­ trast of those segments which, within the experimental time frame, are affected by CLF was matched. Then the dynamics should be equal to those of an asymptotically long, fully con­ fined chain. With the known parameters for PE, eqn [66] yields that, on an average, on each side 220 monomers are released during the observation time of 190 ns. The contrasting experi­ ments were performed on two different chains of molecular weight of 25 kg mol−1, one of which was fully hydrogenated and the other had deuterated labels of about Mw ≅ 4kg mol−1 corresponding to 260 monomers on each end. Both were stu­ died in a deuterated matrix of the same molecular weight.7 In Figure 39, the measured normalized dynamic structure factor S(Q, t)/S(Q) is plotted as a function of time t for different Q values. Figure 39 presents the experimental results for the two chains. Comparing the levels of decay in Figure 39, we realize that S(Q, t)/S(Q) from the fully labeled chain decays significantly stronger than that from the corresponding center-labeled counter part. The constraints are apparently stronger than for the chain, where the ends are visible. We also note that in the case where the ends were masked, the chain center part shows exactly the same structure factor as a very long chain,62 signifying directly the action of CLF at the chain ends and the remaining full confinement in the center.

50

100 t/ns

150

Figure 39 Dynamic structure factor of a center-labeled 25 kg mol−1 PE chain (red symbols) compared to a fully labeled chain (green symbols) of the same overall molecular weight. Q values (in nm−1): 0.5 (squares), 0.96 (circles), 1.15 (triangles). Lines: for center-labeled chain, pure reptation model (eqn [64], red); for fully labeled 25 kg mol−1 chain (green) CLF was considered.7 From Zamponi, M.; Monkenbusch, M.; Willner, L.; et al. Europhys. Lett. 2005, 72, 1039–1044.7

Constraint release

To separate the effect of CR from that of CLF, a labeled chain was considered that was long enough such that CLF did not play a role. The CR effect was investigated in changing the matrix chain length. Thereby the concentration of long chains was low such that overlap effects between the long chains were excluded. The experiment was performed on protonated PE chains (Mw = 36 kg mol−1) mixed into successively shorter deut­ erated chains with molecular weights between 36 and 1 kg mol−1 (the entanglement Mw of PE is Me = 1–2 kg mol−1).63 Figure 40 displays the obtained dynamic structure factors in a Rouse scaling representation of S(Q, t). In the limit QRE ≫ 1 and for times shorter than the Rouse time τR the Rouse dynamic

1.0

h−36 kg mol−1 in d−36 kg mol−1 h−36 kg mol−1 in d−12 kg mol−1 h−36 kg mol−1 in d−6 kg mol−1 h−36 kg mol−1 in d−2 kg mol−1 h−36 kg mol−1 in d−1 kg mol−1

0.8 S(Q, t)/S(Q)

� μðtÞ ¼ 1 −

0

0.6

0.4

0.2

0

0

10

20

30

40

Q 2I 2 Wt Figure 40 Dynamic structure factor of a long labeled chain (Mw 36 kg mol−1) in different shorter matrix chains (Mw = 36, 12, 6, 2, 1 kg mol−1 as indicated in the plot) in a Rouse scaling representation for two different Q values (circles 0.5 nm−1, triangles 1.15 nm−1). For an explanation of the lines, see text. From Zamponi, M.; Wischnewski, A.; Monkenbusch, M.; et al. Phys. Rev. Lett. 2006, 96, 238302.63

Structure Characterization in Fourier Space | Neutron Scattering

structure factor can be approximated by a function depending pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi only on one parameter, the Rouse scaling variable, Q4 Wℓ 4 t 55 if the Rouse approach is valid, then all S(Q, t) need to collapse into one master curve. On the other hand, if topological con­ straints are evident the entanglement length scale comes into play (in the reptation model the tube diameter) and the such scaled dynamic structure factors split up for different Q values. As already demonstrated in Figure 36 for the 36 k chain in a matrix of the same Mw, full confinement is observed. In a Rouse scaling plot as shown in Figure 40, the dynamic structure factor splits for different Q values, displaying the topological con­ straints. The data are well described by de Gennes’ reptation model with a tube diameter as for infinite long chains (lines, eqn [64]). With decreasing matrix length (12–6 kg mol−1), an increas­ ing loss of confinement becomes visible in the form of a stronger decay of the dynamic structure factor. This additional relaxation reflects the phenomenon of CR; the loosening of the tube confinement due to the motion of the surrounding chains. In the scaling representation, the splitting for different Q values is still evident, but the reptation model fails to describe the data. For the 12 kg mol−1 matrix a fit with the reptation model results in a wrong tube diameter, for the 6 kg mol−1 matrix the model also fails qualitatively (dashed lines). Eventually for a short-chain matrix of only one entangle­ ment length (about 2 kg mol−1), the long-labeled chain displays the characteristic Rouse scaling; obviously, the matrix chains are too short to confine the long chain. The data are qualitatively described by the Rouse model (lines), but some slight deviations are visible. For this binary blend, the fitted Rouse parameter is Wl4 = 4.6 nm4 ns−1, which is smaller than the value for long-chain PE Wl4 = 7 nm4 ns−1. Lowering the matrix chain length below the entanglement limit to 1 kg mol−1, the 36 kg mol−1 chain displays undisturbed Rouse motion with a fitted Rouse parameter of close to the expected value Wl4 = 11.3 nm4 ns−1 Wl4 = 12.3 nm4 ns−1 for the short chains when the molecular weight dependence is taken into account.64 Although observing the dynamics of the long chain, the segmental friction of the short chains is measured. Commonly, the reptation time τd of the matrix chains is considered to be the characteristic time scale for CR – a tube forming chain needs to diffuse away in order to release a given chain. For the case of the NSE experiment, we have τd [12 kg mol−1] ≈ 5000 ns and τd [6 kg mol−1] ≈ 500 ns beyond the observation time of the experiment. However, already for the 12 kg mol−1 matrix, which is well entangled, the effect of CR is significant. Thus, other dynamical processes that are known to determine the segmental dynamics also need to be consid­ ered. These processes also include CLF of the chain ends as well as reptational creep.50 In contrast to the center of mass, the segmental mean square displacement is significant in the experimental time range. Using the experimental parameters, an estimate within the Rouse model (eqn [62]) leads to qffiffiffiffiffiffiffiffiffiffi 2 ðτ ¼ 20 nsÞ ¼ ð3:7−4:0Þ nm: rsegm The different dynamic processes were further separated by a pair of experiments: (1) 36 kg mol−1 was studied in a 12 kg mol−1 matrix, and (2) inversely, a 12 kg mol−1 chain was investigated in a 36 kg mol−1 matrix. For a long chain in a short-matrix CLF is negligible; therefore, the effect of CR of

1.0

0.8 S(Q, t )/S(Q )

354

0.6

0.4

0.2

0

0

50

100 t/ns

150

200

Figure 41 Dynamic structure factor of a 36 kg mol−1 chain in a 12 kg mol−1 matrix (solid symbols) and vice versa (open symbols). Q values (in nm−1): squares 0.3, circles 0.5, up-pointing triangles 0.77, diamonds 0.96, and down-pointing triangles 1.15. Lines are just guides for the eye. From Zamponi, M.; Wischnewski, A.; Monkenbusch, M.; et al. Phys. Rev. Lett. 2006, 96, 238302.63

the matrix chains can be observed. On the other hand, for the dynamics of a short chain in a long matrix, the CLF of the short chain are dominating; no CR of the matrix chains can occur. Comparing the dynamic structure factor of such a corre­ sponding pair of samples, the contribution from CLF and CR of the shorter chain can be separated. Figure 41 shows the results. The long-chain relaxation by CR of the short matrix is obviously identical to the relaxation of the confined short chain by CLFs. That is, in the 12 kg mol−1 matrix, the CR visible in the long-chain dynamics may be traced solely to the CLF of the matrix chains. Thus, this experiment shows that even CLF alone can cause the CR effect.63 We may summarize the present status of the NSE experi­ ments on linear polymer melts as follows: in the case of linear chains, the NSE studies have shown that based on the Rouse model the large-scale dynamics may be well understood in terms of topological confinement giving rise to tube constraints as they are suggested by the reptation model. The experimental dynamic structure factors from long chains polymer melts by now rule out all existing competing models to reptation, which so far have produced predictions for S(Q, t). From these results, it is clear that any more fundamental model for polymer dynamics in the melt must contain features of the tube con­ finement as they were phenomenological introduced by the reptation concept. As the leading reptation limiting process at short time, the NSE data have quantitatively confirmed CLFs destroying the tube confinement from the ends. The effect of CR has also been observed directly on a molecular level. It has been experimentally demonstrated that already the CLFs of the chain ends alone may lead to significant CR effects.

2.11.4.4

Soft Confinement

A-B block copolymers show a variety of order disorder phase transitions where the incompatible blocks undergo a micro phase separation. Thus, formed compartments (spheres, cylin­ ders, etc.) implicitly realize a confined space for the inside

355

Structure Characterization in Fourier Space | Neutron Scattering

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 xy ðtÞ ¼ 2=3 tWl 4 =π þ 4DCM t including a small contribu­ tion of the projected center-of-mass diffusion DCM. The undulation effect is contained in 8 9 Qð max >Qðmax 2 > < = 2 � 2 � d Q d Q expð−tγQ=4ηÞ kB T uz ðtÞ ¼ − ½67 > > Q2 2γπ 2 : Q2 ; Qmin

Qmin

The integration limits Qmin and Qmax pertain to the reciprocal dimension of the cylinders (diameter) and the segment size, respectively, γ denotes the surface tension and η the local visc­ osity. Then applying the Gaussian approximation ! 1 X 2� 2 � ~ ~ Q u ðtÞ ½68 IðQ; tÞanisotrop ¼ IðQ; 0Þ exp − 2 j¼x;y;z j j and proper angular averaging � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� � � � � � pffiffiffi � 2 � � erf Q2 2 u2z ðtÞ −2 u2xy ðtÞ π exp −Q2 uxy ðtÞ =2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IðQ; tÞ ¼ IðQÞ � � � � Q 2 u2z ðtÞ −2 u2xy ðtÞ ½69 leads to the necessary expression to describe the bulk-contrast scattering,66 data, and a corresponding fit are shown in Figure 42. Fitting yields a surface tension γ = 4.5 mN m−1 that com­ pares to the prediction using the expression of Helfand and Sapse,67 which yields γ = 3.3 mN m−1. The local viscosity as derived from the Rouse theory implies the involvement of 13 PI units. The Rouse rate also controlling the 2D segmental diffusion corresponds to the homopolymer value. Now this scenario may be applied to interpret the results from the hPI-dPDMS/dPI-dPDMS blend that yields the single-chain PI structure factor, which at first sight is very similar to the single-chain structure factor observed for the homopolymer. However, in the block-copolymer case, one

1.0 0.8 I(Q, t )/I(Q )

0.6 0.4

0.9 0.8

0.2

0.7

f

blocks. Compared to polymer confinement in pores in an anorganic solid (e.g., alumina), the self-assembled boundaries in a polymer system are soft. The degree of ‘softness’ may even be controlled by the incompatibility, for example, employing the temperature dependence of the χ parameter. Using different contrast situations, neutron intensity may be obtained such that different aspects of the dynamics are highlighted. In parti­ cular the h/d contrast in a hA-dB block system will yield intensity that carries the dynamics of the interface between the A and B microphases. On the other hand, a hA-dB/dA-dB blend will show scattering corresponding to the A single chain contrast in the A phase. Besides these basic labelings, others may be envisaged as, for example, dA-hA-dB with a short h-label at the A-B junction which focuses even more to the interface65 or – not yet realized – hA-dA-dB with a label that probes segment dynamics in the inner part of the compart­ ment. To discuss a recent example66 here, we will now concentrate on the system composed by poly(isoprene)­ b-poly(dimethyl siloxane) (PI6-PDMS30, where the numbers indicate the molecular weight in kg mol−1). The corresponding scattering length densities at the measurement temperature of 120 °C are 0.245  1010 cm−2 for h-PI, 6.025  1010 cm−2 for dPI6 (94%D), and 4.545  1010 cm−2 for dPDMS (94%D). This enables the preparation of a hPI-dPDMS/dPI-dPDMS blend (24/76 v/v) such that the average contrast between the PI and the PDMS regions vanishes and the scattering signal is dominated by the single-chain structure factor of PI. SANS on a hPI-dPDMS (‘bulk contrast’) sample shows that the system forms an hexagonal arrangement of long cylindrical PI microdomains. The cylinder radius is 6.4 nm and the unit cell spacing is 27 nm. The scattering intensity from the hdPI-PDMS sample corresponds to the chain conformation and fits to a Debye function. The radius of gyration in the microphase is slightly larger than that in the PI-homopolymer Rg = 2.6 nm versus 2.4 nm indicating that the PI chains in the microphase are slightly stretched. The scattering signal from the ‘bulk contrast’ sample is complex but is essential to understand the role of the inter­ face. In the Q range of NSE experiments (Q = 0.3, …, 1.8 nm−1), the scattering intensity contains – in particular for the lower Q-values – a considerable fraction due to the average form and structure factor of the cylindrical domains. Since these domains are large and therefore slow, this scat­ tering contribution will exhibit no resolvable dynamics within the NSE time window. The other intensity contribu­ tion originates from the A-B intermixing zone close to the interface. The associated contrast between segments on a local scale adds to the intensity the dynamics carried by this signal pertains to the displacement of the ‘blobs’ con­ taining the A-B junction. The main motional contribution for these ‘blobs’ are the interface undulations and the diffu­ sion of the junction point along the interface. The amplitude of undulations depend on the interface tension γ and their relaxation times on the ratio of γ and the local viscosity ηloc. On the other hand, the junction point diffusion resembles the segmental diffusion inside a Rouse chain, however, restricted to 2 of 3 dimensions. To arrive at a viable expres­ sion for the scattering, the starting point is the mean square displacement of the junction blob R2 ðtÞ ¼ u2 xy ðtÞ þ u2z ðtÞ where the in-plane junction segment diffusion contributes

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Figure 42 NSE results showing the intermediate dynamic structure factor from the hPI-dPDMS bulk contrast at Q = 0.8, 1.1, 1.3, 1.5, and 1.8 nm−1. The solid lines display fits from the model taking into account interfacial undulations and Rouse segmental diffusion in addition to the static contribution from the whole cylinders. The inset shows the fraction, f, of inelastic contribution as a function of Q. The dashed lines display the dynamic structure factor without the interface undulation contribution. Reproduced with permission from Willner, L.; Lund, R.; Monkenbusch, M.; et al. Soft Matter 2010, 6, 1559–1570.66

356

Structure Characterization in Fourier Space | Neutron Scattering

must assume that the PI chain is fixed to the junction point and any model description has to start with a Rouse chain tethered at one end. From the bulk-contrast scattering, the undulation and diffusion of the tethering point is known. The only addi­ tional assumption pertains the extension of the extra dynamics of the junction point along the chain into the inner part of the micelle. All the PI-PDMS data show that the overall Rouse dynamics prevails, although it is affected and accelerated by interface undulations, which contribute mainly at high Q and counter­ acts in parts the effect of grafting one chain-end to the surface. Secondly, at lower Q, in agreement with other studies, the diffusion is essentially slowed down by a confinement to the interface. This diffusion can be estimated remarkably well based on the Rouse theory and Fickian diffusion in 2D. Hence, while the overall global dynamics slows down, the chain dynamics is still similar to Rouse motion.

2.11.4.5

Dynamically Asymmetric Blends

Blends of different polymers with very different glass transition temperatures Tg are expected to exhibit new dynamical effects due to the close interpenetration of chains with a high variation of mobility. This may be utilized to adjust the mechanical prop­ erties. However, the underlying combination rules of mixing have to be understood to fully exploit the blending method. Among the known miscible polymer systems (see, e.g., introduction of Reference 68 for a compilation) PEO/poly (methylmethacrylate) (PMMA) is among the best investigated ones and exhibits a large dynamical asymmetry as expressed by the huge difference of the glass transition temperatures Tg (PEO) ≅ 200 K and Tg(PMMA) ≅ 400 K. Careful calorimetric measurements reveal the occurrence of two glass transitions in the mixtures.68,69 These findings are rationalized using the self-concentration concept of Lodge and McLeish70 which takes into account that the chain connectivity leads to an enhanced probability to find a segment of the same chain close to a given segment. The volume of influence approximately has the size of the Kuhn length lK. of the polymer. Thus, an estimate for the self-concentration is Φs ¼

C∞ M0 nρNA V

½70

where C∞ is the characteristic ratio, M0 the monomer molecular weight, n the number of bonds per repeat unit, ρ the density, NA Avogadros number, and V the effective Ð 3 volume lK . With the effective volume of polymer component a Φaeff ¼ Φas þ ð1− Φas ÞΦa the effective glass temperature may be predicted using the Fox equation71 " # Φ ð1− ΦÞ eff Tg ðΦÞ ¼ Tg ðΦeff Þ; Tg ðΦÞ ¼ a þ ½71 Tg Tgb The calorimetric observation of two class transitions and their dependence on composition can be reasonably well explained by the above Ansatz. This is partly true also for the dynamic features observed by inelastic neutron scattering on PEO/ PMMA.72 Incoherent scattering measured by direct time­ of-flight neutron spectroscopy reveals two relaxation processes in the ps range and on the monomeric length scale. The fast process found is insensitive to mixing and resembles one of the

pure polymer, whereas the characteristic times of the slow process follow a Vogel–Fulcher relation if the Vogel–Fulcher temperature is changed by the Tg difference according to the self-concentration inserted into the Fox equation as above. This relation explains data pertaining a spatial scale from 7 to 18 Å. Up to now PEO/PMMA is the only highly asymmetric blend that has been investigated by neutron time of flight (TOF), backscattering, and NSE spectroscopy.73 The experiments aiming at the incoherent scattering of PEO protons (TOF and back scattering spectrometer (BSS)) were performed with hPEO/dPMMA samples with 25%, 35%, and 50% PEO. The NSE experiments aimed at the Rouse dynamics as seen from the single-chain structure factor; therefore, the PEO fraction was prepared as a mixture of hPEO and dPEO. Experiments were conducted at temperatures 350, 375, and 400 K. The glass transition of the fast PEO component is about 200 K. The back­ scattering results on local (segmental) diffusion of PEO pffiffi exhibits a mean square displacement that increases ∝ t as expected for a Rouse chain. The rate, however, is significantly reduced compared to pure PEO, for 35% PEO and 400 K the reduction factor is 6. The Fourier transform of the TOF data explores the time between fractions of a ps to several 10 ps and may be combined with BSS data extending to some 1000 ps, revealing that the local sub-ps dynamics of PEO is not affected by PMMA blending, but the intermediate regime from several ps to ns exhibits slowing down and stretching. The latter indi­ cates a broad distribution of local friction coefficients. The significance of the TOF and BSS data analysis was enhanced by the exact knowledge of the ratio of coherent and (spin) incoherent scattering intensity, which was obtained by neutron diffraction with polarization analysis. The coherent scattering of hPEO/dPMMA carries the frozen dynamics of the PMMA component. The low Q coherent scattering intensity analyzed in the NSE experiments on hPEO/dPMMA exhibit no dynamics since the scattering length density modulation due to PEO versus PMMA contrast can only change if the PEO and the PMMA component move. The latter is frozen and motion is too slow to resolve within the NSE time window of several 100 ns. Therefore, in order to investigate the PEO dynamics, the PEO component has to be a mixture of H- and D-chains. This contrast yields scattering intensity associated with the PEO single-chain con­ figuration and motion within the PEO regions. Still there is an elastic contribution from the immobile PMMA contrast; it can be quantified by the RPA Ansatz. In the NSE data, it is seen as a constant offset in the normalized intermediate scattering func­ tion. Figure 43 displays results obtained at the IN15 NSE spectrometer at the ILL Grenoble. On top of the constant frac­ tion that follows from the RPA analysis of scattering associated to the dPMMA contrast versus the dhPEO fraction, relaxation corresponding to the single-chain structure factor of PEO is seen. This dynamical contribution is more stretched and lags significantly behind the prediction of the simple Rouse model (lines); however, it still exhibits Rouse scaling. The latter indi­ cates that there is no additional intermediate length scale as, for example, the tube diameter in entangled systems or a length scale imposed by structures of the PMMA matrix that can be associated to the observed dynamics. The decay for the three lower Q values can perfectly be described by a modified Rouse model with a random distribu­ tion of bead friction coefficient instead of a uniform friction.

Structure Characterization in Fourier Space | Neutron Scattering

show no evidence for a characteristic length scale, which would indicate confinement or random obstacles. With a random distribution of friction coefficients in the Rouse model, it is possible to quantitatively describe both the local and the dynamics at the chain level with a common set of parameters. The apparent strong disagreement of the dynamics at the chain level with that from macroscopic diffu­ sion is not clear, but might be related to the broad distribution of mobilities, mesoscopic structure formation, or entropic trapping or to all of them.

1.0 Q = 1 nm−1 Q = 1.5 nm−1 Q = 2 nm−1 Q = 3 nm−1

S(Q, t )/S(Q )

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2.11.5 Conclusions 0.0 0

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40

Q 2I 2(Wt )1/2 Figure 43 Symbols indicate NSE data from hPEO/dPEO/dPMMA at 400K for different Q values displayed as function of the Rouse scaling variable. The dashed region indicates the elastic contribution from the frozen dPMMA contrast. The solid lines are the predictions of the simple Rouse theory. Reproduced with permission from Niedzwiedz, K.; Wischnewski, A.; Monkenbusch, M.; et al. Phys. Rev. Lett. 2007, 98, 168301.73

Assuming a log–normal distribution of friction coefficients introduces one new parameter, the distribution width σ. Fitting yields σ = 1.6. With that value, the decay of the inter­ mediate scattering function for the three lowest Q values can be perfectly reproduced. The predicted relaxation for Q = 3 nm−1 is somewhat more pronounced than observed.73 The log–normal distribution with σ = 1.6 describes the broad distribution of local relaxations as observed by the TOF and BSS experiments without further parameters as depicted in Figure 44. In summary, for short times the self-dynamics of PEO follows the Rouse expectation, however, with a significant larger friction coefficient compared to pure PEO. The collec­ tive chain dynamics at longer times are further retarded and 0.8 pure PEO 35% PEO/PMMA (TOF) 35% PEO/PMMA (BSS)

Ssdf(Q, t )

0.6

T = 400 K Q = 19 nm−1

0.4

0.2

0.0 0.1

1

10 Time (ps)

100

1000

Figure 44 Fourier transform of TOF and backscattering (BSS) data from pure PEO and the PEO/PMMA blend. The solid line through the pure PEO data corresponds to a stretched exponential with stretching exponent β = 0.5. The line through the blend data is derived with the log–normal friction distribution with the width parameter as obtained by NSE. Reproduced with permission from Niedzwiedz, K.; Wischnewski, A.; Monkenbusch, M.; et al. Phys. Rev. Lett. 2007, 98, 168301.73

In this chapter, we introduced the techniques of SANS and NSE spectroscopy and demonstrated that both techniques are powerful tools to investigate structures, interactions, and mobi­ lities in polymer systems. The importance of these methods mainly arises from the unique ability to distinguish chemically equal items within a sample by H/D isotope labeling. Starting with seminal SANS experiments that showed the Gaussian coil nature of linear polymer chains in melts, which was only possible by labeling a few equal chains and exploiting the large difference in scattering contrast for neutrons. Meanwhile beyond those, more complicated partial labeling schemes can and have been employed to highlight certain parts of a polymer chain. As an example pertaining to the dynamics (NSE), a special chain-end labeling was applied to mask the effects of CLFs (Section 2.11.4.3) and to focus on different structures in block-copolymer micelles (Section 2.11.4.4). Polymer(H) solu­ tions in deuterated solvents yield a large neutron scattering contrast compared with X-ray scattering that covers the same spatial range. Routinely obtained absolute intensities further provide vital information about molecular parameters and discriminate models. Neutrons are a ‘gentle’ probe that causes virtually no radiation damage in the samples, a feature that partly depends on the weak interaction and limited flux of neutron beams requiring, on the other hand, a large volume of the sample. SANS provides a unique tool to explore structural details of polymeric systems and allows kinetic studies with a resolution of a few seconds or less. Phenomena observed in polymer blends allow molecular interactions to be assessed as well as proof of theoretical predictions of static and kinetic properties of, respectively, critical phenomena and phase transition (Sections 2.11.3.2 and 2.11.3.3). Quenched SANS allows the evolution of anisotropic scattering patterns after application of sudden strain thereby extending the effective time window for the observation of polymer relaxation and their hierarchy in systems with more complicated architectures from spin echo toward macroscopic times using time–temperature scaling (Section 2.11.3.4). The ability to observe the mobility of nm large structures in a polymer sample by NSE relies on the same labeling and is unique in its space–time coverage leaving only a small gap to the realm of dynamic light scattering on the low Q side. NSE probes the mobility of segments and larger structures. Special H/D labeling helps to distinguish effects due to interface tension from Rouse-type motions within a bulk polymer compartment. Dynamical tube confinement as the main ingre­ dient of reptation theories can be scrutinized in detail. Polymer

358

Structure Characterization in Fourier Space | Neutron Scattering

chain/segments can be explored in a time window from a few ps to several 100 ns a feature that may not show any scattering signature in ‘static’ scattering experiment. In summary, the presented neutron scattering methods – eventually with others like reflectometry, grazing incidence, and large angle diffraction – which in combination with chemistry, and preparing well-defined (partially) isotope-labeled polymers, provide a unique view on struc­ tures and interactions in these nano- and mesoscopic soft-matter systems.

Appendix Following Figure 11, we now analyze in detailed the NSE instrument and the connection of measured intensities with the neutron scattering function. The precession angle before and after, the sample and the π-flipper, respectively, is propor­ tional to the time the neutron stays in the precession magnetic field. Each infinitesimal section of the path Δℓ hosts the neu­ tron during Δt = Δℓ/ν with the neutron velocity ν = (h/mn)λ−1 and thereby contributes ΔΨ ¼ γjBðlÞj Δt with the Larmor con­ stant γ = 2π  2913.06598  104 rad s−1T−1 and B(ℓ) the local magnetic field (induction). The full path (I = 1, 2) then contributes Ψi ¼

ð li ðπÞ li ðπ=2Þ

ΔΨ dl ¼

γ ðh=mn Þλ−1

ð li ðπÞ

� � �BðlÞ� dl li ðπ=2Þ |fflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflffl}

½A1

J1;2

The action of the π-flipper rotates the neutron spin ensemble such that effectively Ψ1 → –Ψ1. Inserting the resulting total precession angle at the end of the second precession track Ψ = –Ψ1 + Ψ2 into the analyzer transmission T and integration over the neutron velocity distribution after scattering according to the spectral function S(Q, ω) yields the detector intensity � ð� 1 γJ1 IDet ∝ 1  cos − 2 ðh=mn Þλ−1 �� γJ2 þ SðQ; ωÞdω ½A2 ðh=mn Þλ−1 þ λω=2π Observation that ΔE = ℏω causes an increment Δν = λω/2τ of the neutron velocity in the second arm and a little algebra then yields � � � 1 γðJ2 −J1 Þλ IDet ∝ SðQÞ  cos 2 ðh=mn Þ ! � ð 3 γJ2 λ  cos ω SðQ; ωÞdω ½A3 2πðh=mn Þ2 Ð where S(Q) = S(Q, 0) = S(Q, ω)dω. Since J1 ≈ J2 = J, the further results are expressed in terms of J and ΔJ = J2 – J1. � � � ð ð 1 mn IDet ðJ; δ; λ0 Þ∝ wðλ − λ0 Þ SðQÞ  η Wðδ − ΔJÞcos δγ λ 2 h � � ð � 2 m  cos Jλ3 γ n 2 ω SðQ; ωÞdωdδ dλ 2πh ½A4 In a real spectrometer, the incoming wavelength is not arbitrarily well defined, but stems from a distribution w(λ – λ0), which we approximate by 1/(Λπ1/2) exp(–[(λ – λ0)/2Λ]2). At a reactor-based

spectrometer, this reasonably well describes the velocity distribu­ tion from a mechanical velocity selector. Also the field integral difference ΔJ has a certain variance which is due to slightly differ­ ent values for different paths pairs from π/2-flipper to the scattering point in the sample and then to the final π/2-flipper, which are members of the path ensemble that connects (via scattering) the source area with the effective detection area. For simplicity, here also this distribution is approximated by a P P Gaussian: W(δ – ΔJ) ≈ 1/( π1/2) exp(–[(δ – ΔJ)/2 ]2). where δ denotes the nominal field integral asymmetry. A parameter η ≈ 1 introduced to account for imperfect polarization analysis and possible depolarization effects. Then the detector intensity is " � X � � � ð 2 1 mn 2 2 mn 2 γ 2 λ cos δγ λ ¼ wðλ−λ0 Þ SðQÞ  η exp − h 2 h # � � ð m2  cos Jλ3 γ n 2 ω SðQ; ωÞdω dλ ½A5 2πh P where R = ηexp(– 2γ2(mn/h)2 λ2) approximates what is usually considered as the resolution function of an NSE spectrometer. Note that – at high resolution – generally ∑2  J2. To proceed further, we abbreviate the Fourier time t = Jλ3γ(mn/h)2/2π = Cλ3, t0 ¼ Cλ0 3 , and write Q = 4π/λsin (θ/2) = q/λ and expand the cos-Fourier integral as a function of λ around the nominal wavelength λ0 cosðωCλ3 ÞSðq=λ; ωÞ ¼ cosðωt0 ÞSðq=λ0 ; ωÞ � d ðλ−λ0 Þ − cosðωt0 ÞQ0 SðQ0 ; ωÞ dQ0 λ0 � þ3sinðωt0 Þωt0 SðQ0 ; ωÞ þ Oπðλ2 Þ ½A6 integration over ω then yields SðQ; tÞ ¼ SðQ0 ; t0 Þ− −3t0

� d ðλ−λ0 Þ Q0 SðQ0 ; t0 Þ dQ0 λ0

d SðQ0 ; t0 Þ þ Oðλ2 Þ dt0

½A7

The quadratic expansion term in its final form is " d2 d 1 ðλ−λ0 Þ2 SðQ0 ; t0 Þ þ 2Q0 SðQ0 ; t0 Þ Q20 2 2 λ0 dQ0 dQ20 d2 d SðQ0 ; t0 Þ þ 6t0 SðQ0 ; t0 Þ dt0 dt02 # d2 −6Q0 t0 SðQ0 ; t0 Þ dQ0 dt0 þ9t02

½A8

The λ-expansion of the cos-Fourier integration allows now to include the implicit λ-dependence of SðQ; tÞ into the analysis of the effects of finite width wavelength and field integral distributions. Without consideration of this implicit depen­ dence, we get 2 39 8 > > 2 > > 1 m > 2 2 2 2 2 2 n > > SðQÞ  η exp4−ðΣ λ0 þ Λ δ Þγ 2 =A 5 > > > > > < = A h 1 0 1 I∝ ½A9 > > 2> > mn > > 2A > > @ >  cos δγ λ0 =A SðQ0 ; t0 Þ þ ⋯ > > > : ; h

Structure Characterization in Fourier Space | Neutron Scattering

where A ¼ 1 þ 4Σ2 Λ2 γ2

�m �2 n

h

½A10

To arrive at a more compact form, we now introduce the abbreviations g = γ(mn/h) = 4.627  1014 T−1 m−2, m2

Ψ2 ¼ ½Σ2 λ20 þ Λ2 δ2 γ2 h2n =A2 and Φ ¼ δ γ mhn λ0 =A2 well as the expansion coefficients S0 = S(Q0, S1 ¼ −Q0 dQd 0 SðQ0 ; t0 Þ þ 3t0 dtd0 SðQ0 ; t0 Þ; S2 ⋯ :

as t0),

Combining terms with equal power of Λ, then yields

I∝

" � 1 1 2 Λ2 SðQÞ  η expð−ψ2 Þ cosðϕÞS0 − 2 2 2 A A λ0 �� �  A2 ϕ sinðϕÞ þ 2ðΣλ0 gÞ2 cosðϕÞ S1 �# � 1 − 2 cosðϕÞS2 þ ⋯ 2A

½A11

To assess the importance of the higher terms in the expansion, the parameters have to be related to realistic values as obtained in existing NSE experiments. First, ∑ is related to the mean square path integral difference 〈ΔJ2〉/2 = ∑2 and Λ is related to the FWHM Δλ of the wavelength distribution by Λ = Δλ/(4ℓn2)1/2) = Δλ/3.33. Thus, for example, for λ0 = 1 nm, Δλ = 0.1 nm (Λ = 0.03 nm), ∑ = 2  10−6 Tm, we obtain A = 1.0015, ∑λ0g = 0.9254 (Λ/Λ0)2 = 0.032 = 0.0009 and R/η = 0.43. Since the expansion parameter (Λ/λ0)2 is typically small (10−3), the higher orders in eqn A11 are negligible for all cases with reasonably smooth behavior of S(Q, t) as expressed ðnþmÞ

∂ SðQ; tÞ Q t << 1 for n, m = 0, 1, 2, …. SðQ; tÞ ∂n Q∂m t This is the explanation for the observation that the intermediate scattering functions which are typical for simple polymer systems do not suffer from significant distortions due to the typical large incoming neutron wavelength spread. To measure the intermediate scattering function at a given setting of (Q, t), that is, scattering angle, nominal wavelength,

by ðΔ= λÞ2ðnþmÞ

n m

359

and magnetic precession field, the detector intensity as function of symmetry is scanned. This procedure ensures that the infor­ mation of the exact location of the symmetry point is also obtained. An example is shown in Figure A1, the Gaussian type envelope of the echo signal reflects the finite wavelength spread. The information on the normalized intermediate scattering function is given by the ratio of the oscillation amplitude and the limiting intensities (Iup – Idown). The amplitude a = R S(Q, t) contains information on the instrumental resolution R ≅ ηðλ0 ; tÞ A1 expð−Σ2 λ20 Þ and S(Q, t) and can be extracted from the results of the symmetry scan � � � � m2 � 1 IðδÞ ¼ ε SðQÞ  a exp − Λ2 fδ − δ0 g2 γ2 2n =A2 2 h � �� mn cos fδ − δ0 gγ ½A12 λ0 =A2 h In the vicinity of the true symmetry point, the analysis may be based on 3 or 4 points and neglecting the envelope due to finite wavelength width. Alternatively, fitting of eqn [A12] may be used to determine a, δ0, and the average value S(Q). Usually, A ≅ 1 is assumed. The intensities Iup and Idown are determined with the same setting with either none or only the π-flipper operating and represent S(Q) = (Iup – Idown)/2ε. Then, the nor­ malized result is obtained as 2ε a SðQ; tÞ ¼ RðQ; λ0 ; tÞ SðQÞ Iup − Idown

½A13

to correct for the resolution the factor R(Q, λ0, t) is determined by a scan using a reference sample that scatters purely elasti­ cally, that is, S(Q, t)/S(Q) ≡ 1. Finally, the resolution correction consists of a simple division. Using wavelengths λ between 0.2 and 2 nm and J = 0.001, …, 1 Tm, a time range from 1 ps to 1 μs is acces­ sible. However, the lowest usable Q value of > 0.1 nm−1 is still 1 order of magnitude larger than the lowest wave vector accessible by conventional SANS and more than 2 orders

10000 10000 8000

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Symmetry (phase current (40 turns)/A)

Figure A1 Symmetry scan showing the echo signal as obtained with a wavelength spread of 10% FWHM. The symmetry point is at about 1A phase current. The dash–dotted lines indicate the intensity limits: low (Idown) for the blocking combination of analyzer and π-flipper, high (Iup) for the transmitting combination. The echo amplitude may be obtained by nonlinear fitting for an arbitrary number of points or close to the symmetry point by direct calculation from 3 (minimum) or 4 points (red diamonds) with fixed known phase angle step (here 90 °).

360

Structure Characterization in Fourier Space | Neutron Scattering

larger than those of U-SANS techniques. The limitation is, on the one hand, given by the NSE beam divergence and the increasing parasitic small-angle scattering from in-beam com­ ponents as flippers and correction coils, on the other hand, even the largest Fourier times are too short to detect the slow very large-scale motions seen at the lowest Q values, the characteristic times increase like τ ∝ Q−2 … −4. Further descriptions of the NSE technique and instruments can be found in references.16

References 1. Squires, G. L. Introduction to the Theory of Thermal Neutron Scattering; Cambridge University Press: Cambridge, 1978. 2. Lovesey, S. W. Theory of Neutron Scattering from Condensed Matter”

Clarendon Press: Oxford, 1987.

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Biographical Sketches Dieter Richter studied physics at the Technical University Braunschweig and the RWTH-Aachen, where he obtained his PhD. After a postdoctoral at the Brookhaven National Laboratory under G. Shirane, a Habilitation in Aachen and a 5-year term as senior scientist and group leader at the ILL in Grenoble in 1989, he became director at the IFF in Jülich and professor at the University of Münster. His main research areas are soft matter science and neutron scattering techniques. In 1994, he founded the European Neutron Scattering Association and assumed leading positions in various research projects and organizations. For his scientific work, he has received a number of important research awards.

Michael Monkenbusch studied physics at the University of Münster and finished with a thesis work on neutron scattering on absorbed molecules. After 4 years at the Institute of Macromolecular Chemistry in Freiburg working on electrically conducting polymers, he entered the Institute for Solid State Research (IFF) in Jülich as a researcher. There he was responsible for building the neutron spin echo (NSE) spectrometer and its exploitation for a variety of soft matter problems from polymer melt dynamics to protein motion.

Dietmar Schwahn studied electrical engineering at the RWTH-Aachen and performed his PhD in physics at the University of Bochum. Since then, he has been a member of the Institute of Solid State Division (IFF) of FZ-Jülich and became the responsible scientist for the small-angle neutron scattering (SANS) instruments. He was the project leader for the design and construction of the present pinhole small-angle instruments KWS1 and KWS2 and the very high-resolution double crystal diffractometer (DCD). For a long time, he was engaged in the field of critical phenomena and phase transition in polymer melts and shifted some years ago to the field of biomineralization.