1

Small Angle Neutron Scattering on biological and synthetic molecules Manuscript for experiments on the KWS-2 by Henrich Frielinghaus / M.S. Appavou The objective of this lab course is to clarify the essential concepts of small-angle neutron scattering. Structures are only visible by a scattering experiment if there is an appropriate contrast. For neutrons one often uses the exchange of 1H by 2H, i.e. Deuterium. In this manuscript, the term ‘scattering length density’ will be derived to be the fundamental magnitude of a small angle scattering experiment. We will investigate the protein Lysozyme in heavy water (Fig. A) to obtain the information about the typical distance of the molecules. A second example is a micellar solution of the amphiphilic polymer-POO10 PEO10 in heavy water (Fig. B). The chosen contrast makes only the core of the micelle visible. The core is very compact because of the hydrophobic interaction. The aim of this second part is to determine the radius of the cylindrical structure. The theoretical concepts of small-angle neutron scattering are derived in detail, but without too many formulas to obtain a vivid picture.

Figure A: Representation of the protein lysozyme, which has a very compact form.

Figure B: Schematic representation of a cylindrical micelle, which is composed of amphiphilic polymers.

2

Basics of small-angle neutron scattering 1. Principle and instrumental technique At the research reactor FRM2 in Garching, the neutron radiation is used for experiments. In many cases, materials are examined in terms of structure and dynamics. The word neutron radiation already contains the wave-particle duality, which can be treated theoretically in quantum mechanics. By neutron we mean a corpuscle usually necessary for the construction of heavier nuclei. The particle properties of the neutron become visible when classical trajectories are describing the movement. The equivalent of light is obtained in geometrical optics, where light rays are described by simple lines, and are eventually refracted at interfaces. However, for neutrons the often neglected gravity becomes important. A neutron at a (DeBroglie) wavelength of 7Å (=7x10-10m) has a velocity of v = h / (mn∙λ) = 565 m/s. At a length of 20m this neutron is therefore falling by 6.1 mm. Thus, the design of neutron instruments is oriented to straight lines with small gravity corrections. Only very slow neutrons show significant effects of gravitation, such as the experiment of H. Meier-Leibnitz described at the Subway station ‘Garching Forschungszentrum’. The wave properties of neutrons emerge when there is an interaction with materials and the structural size is similar to the neutron wavelength. For the neutron wavelength 7Å these are about 5 atomic distances of carbon. For a Small Angle Neutron Scattering (SANS) experiment we will see that the typical structural sizes investigated are in the range of 20 to 3000Å. The coherence of the neutron must, therefore, be sufficient to examine these structural dimensions. Classically this consideration will be discussed in terms of resolution (see below). The scattering process appears only due to the wave properties of the neutron. A scattering experiment is divided into three parts. First, the neutrons are prepared with regard to wavelength and beam alignment. The intensity in neutron experiments is much lower than in experiments with laser radiation or X-rays at the synchrotron. Therefore, an entire wavelength band is used, and the divergence of the beam is limited only as much as necessary. The prepared beam penetrates the sample, and is (partly) scattered. For small angle neutron scattering experiments, the scattering process is elastic, i.e. there is virtually no energy transferred to the neutron. However, the direction changes in the scattering

process. The mean wave vector of the prepared beam ik

(with /2ik

) is deflected

according to the scattering process to the final wave vector fk

. The scattered neutrons are

detected with an area detector. The experimental information is the measured intensity as a function of the solid angle Ω. This solid angle is defined relatively to an ideally small sample and for large detector distances. In practice, the classical small-angle neutron scattering apparatus including the source looks like this: In the reactor a nuclear chain reaction takes place. A uranium nucleus 235U captures a free neutron, and fission to smaller nuclei takes place. Additionally, 2.5 neutrons (on average) are released, which are slowed down to thermal energy by the moderator. One part of the neutrons keeps the chain reaction going on, while the remaining part can be used for the neutron experiments. The cold source is another moderator, which cools the neutrons to about 30K. Here, materials with light nuclei (deuterium on FRM2) are used to facilitate the thermalization. The cold neutrons can easily be transported to the instruments by neutron guides. Rectangular glass tubes are used with a special mirror inside. The neutron

3

velocity selector works mechanically (Fig. 1 shows scheme). A rotating cylinder with tilted lamellae allows only neutrons with a certain speed to pass . The wavelengths distribution is ideally triangular with a relative half-width of ± 5% or ± 10%. The collimation determines the divergence of the beam. The entrance aperture and the sample aperture have a distance LC, and restrict the divergence of the beam. The sample is placed directly behind the sample aperture. Many unscattered neutrons leave the sample and will be blocked by an absorber in front of the detector. Only the scattered neutrons are detected by the detector at a distance LD. The sensitive detector detects about 93% of the scattered neutrons, but the huge primary beam cannot be handled. In the apparatuses KWS-1 and KWS-2 the beam stop contains a small counter to measure the unscattered neutrons in parallel. This classic small-angle neutron scattering apparatus is also known as pinhole camera, because the entrance aperture is imaged to the detector by the sample aperture. The sample aperture may be opened further if focusing elements maintain (or improve) the quality of the image of the entrance aperture. By focusing elements the intensity of the experiment may be increased on the expense of larges samples. Focusing elements can be either curved mirrors or neutron lenses made of MgF2. Both machines KWS-1 and KWS-2 have neutron lenses, but throughout this lab course they will not be used.

Figure 1: Principle of a small-angle neutron scattering diffractometer. The neutrons pass through the neutron velocity selector. The divergence is determined by the two apertures (entrance and sample aperture of the collimation). Then, the neutrons penetrate the sample. The unscattered neutrons are absorbed at the beam stop, while the scattered neutrons are detected by the position sensitive detector. 2. The scattering vector Q

In this section, the scattering vector Q

is described with its experimental uncertainty. The

scattering process is schematically shown in Fig. 3, in real space and momentum space. In real space the beam hits the sample with a distribution of velocities (magnitude and direction). The neutron speed is connected to the wavelength, which distribution is depending on the velocity selector. The directional distribution is defined by the collimation. After the scattering process, the direction of the neutron is changed, but the principal inaccuracy remains the same. The scattering angle 2Θ is the azimuth angle. The remaining polar angle is not discussed further here. For samples with no preferred direction the

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Figure 2: Setup of the SANS instrument KWS-2 in the neutron guide hall of FRM-2, Garching. In the middle of the picture is the sample position (A collaborator mounts a sample). The neutrons are coming from the left through the collimation. On the right is the detector tank with its entry window. The neutrons are transported largely in vacuum due to absorption by air.

Δk/k

F

Sample

2

2

ΔΩ, Δλ/λ

ΔΩ, Δλ/λ

ΔΩ

ΔΩ

Q

ki

kf

Δk/k

F

Sample

2

2

ΔΩ, Δλ/λ

ΔΩ, Δλ/λ

ΔΩ

ΔΩ

Q

ki

kf

Figure 3: Above: the neutron speed and its distribution in real space, before and after the scattering process. Bottom: The same image expressed by wave vectors (reciprocal space). The scattering vector is the difference between the outgoing and incoming wave vector. scattering is isotropic and, thus, does not depend on the polar angle. For the samples we use here, we can safely assume isotropy. In reciprocal space the neutrons are defined by the

wave vector k

. The main direction of the incident beam is the z-direction, and the modulus

is determined by the wavelength, so /2ik

. Again, k

is distributed due to the selector

and the collimation. The wave vector of the (quasi) elastic scattering process has the same

5

modulus, but differs in direction, namely by the angle 2Θ. The difference between both wave vectors is given by the following modulus:

(1)

For isotropic scattering samples, the measured intensity depends only on the scattering vector Q. For small angles, the common approximation of small angle neutron scattering is valid:

(2)

The typical Q-range of a small angle scattering apparatus thus follows from the geometry. The detector distances LD vary in the range from 1.5 m to 20m. The area detector is active between dD = 4cm and 35cm from the center. The angle 2Θ is approximated by the ratio dD/ LD and the wavelength λ is 7Å. For the apparatuses KWS-1 and KWS-2, a typical Q-range from 2∙10-3 to 0.2 Å -1 is obtained. The Q-vector describes which length scales ℓ are observed, following the rule ℓ = 2π/Q. If a Bragg peak is observed, the lattice parameters can be read of from the position of the peak. If the scattering law shows a sudden change at Q, we obtain the length scale of the structural differences. Scattering laws may be simple power laws Qα with different exponents α. 3. The macroscopic scattering cross section In this section we will learn how the scattering intensity is theoretically described. The neutron, because it does not carry a charge, mainly interacts with the atoms of the sample due to nuclear forces. In principle, also the magnetic moment of the neutron interacts with magnetic fields, which are caused mainly by the unpaired electrons. But this experiment focuses on soft matter samples which consist of organic materials and which therefore are non-magnetic. A single nucleus is much smaller than the wavelength of the neutron (about 10-4Å in comparison to 7Å). Therefore, the scattered waves of single nuclei are only s-waves. These are spherical waves, which have the same intensity in any direction. The relative phase can vary between 0 and π. The phase 0 is true for most of the nuclei, while the latter is found, for example, for the simple hydrogen nucleus 1H (contrarily to 2H, the deuterium D). This important difference between the two hydrogen nuclei is used in the field of soft matter research, to highlight certain molecules or parts of the investigated molecule. The scattering probability of a single atom is given in terms of an area, the scattering cross section. This area is simply the square of the scattering length b. The ratio of the scattering cross section and the illuminated area is the probability for the scattering process, or in the quantum mechanical sense the relative intensity of the scattered waves. The relative phase of π results just in a negative scattering length. In principle, absorption can lead to a complex scattering cross section. For simplicity, we will neglect this detail. A real sample of condensed matter consists of many atoms. Each of these atoms emits individual spherical waves according to the scattering length, which interfere and finally are detected by the detector. This description corresponds to Huygens' principle. On the one hand, the amplitude of the individual spherical waves decreases with a 1/r law (the integrated intensity of each sphere is therefore constant). On the other hand however, since

6

the detector is found at large distances, this 1/r law becomes infinitely weak for the small distances of different atoms. More important is the optical path difference of the individual atoms for the phase of the outgoing wave. This leads to interference effects. The outgoing amplitude A of all atoms can be written as:

(3)

The scattering lengths bj are unique for each nucleus j on the location rj. The complex unit is denoted by i. While A is the amplitude, the macroscopic scattering cross section dΣ/dΩ as a function of the scattering vector Q is proportional to the intensity:

(4)

The macroscopic scattering cross section is further normalized by the volume of the sample, such that it is simply proportional to the total number of scattering centers or the volume of the sample. In principle, the aim of this section is reached with this formula. However, this formula can be reformulated for different interpretations. In detail, some effects of small angle neutron scattering are discussed. The typical Q range of small angle neutron scattering covers length scales of about 30 to 3000Å and thus concentrates on structures that already contain many atoms. The description of isolated atomic scattering centers, as in Equation 3, contains more details than a small angle scattering experiment can resolve. This means that, for this angular range, the description of the interfering waves can be approximated. So, several atoms can be combined. A low molecular weight organic substance may be described by the sum of all Σbj scattering lengths and the occupied volume vmol (molecular volume). The quotient is the scattering length density ρ = Σbj / vmol. For large molecules this concept has to be applied for subsections. Usually one considers the monomers, which are the building blocks of polymers. In this sense, equation 3 looks like this:

(5)

The individual substances are now considered to be smeared homogeneous volumes that fill the entire sample volume. On larger length scales this is the simplified representation. Are, for instance, domains formed by small molecules, only the entire domain is visible by small angle neutron scattering. The domains can be regarded as homogeneous volumes, without regarding the details of all the individual atoms. The contrast for neutrons is produced mostly by chemical deuteration. The interaction between deuterated and protonated chemicals often do not differ greatly from the purely protonated substances. The use of D2O in microemulsions shifts the phase boundary down by 2K. In polymers, however, this shift might result in many 10K, due to the sum of all monomers in a high molecular weight polymer.

So far we have neglected the nuclear magnetic moment, because it is not oriented. But nevertheless there exists a statistics of the relative orientation of nuclear moments with respect to the magnetic moments of the incident neutrons, which are often not polarized. We consider a nucleus having a spin ½ like the hydrogen atom. The neutron also has a spin ½ . Now, the relative orientation can be antiparallel or parallel. Specifically, the total spin S is

7

0 (singlet) or 1 (triplet state). For both states, there is an individual scattering length. Statistically, only the mean scattering length and the variance of the scattering length are important. In average, the nucleus scatters with an amplitude of the mean scattering length. This scattering phenomenon is called coherent scattering, and the corresponding scattering length is the coherent scattering length. With this coherent scattering length, the interferences are considered to be coherent between all atoms (i.e. the relative phase is exp(-iQr)). The fluctuations of the individual scattering lengths are related to the variance. The incoherent scattering results from individual scattering nuclei, which have no fixed phase relationship. Each of these nuclei behaves as an individual point scatterer, and the incoherent scattering is independent of the wave vector Q, i.e. a spherical wave. Both of these features appear as the sum:

(6)

For the area of soft matter (i.e. mostly organic molecules) the hydrogen 1H is the most important atom, which causes incoherent scattering. At too high amounts of 1H hydrogen the incoherent scattering hides important structural information of the coherent scattering. Therefore, the component with the highest volume fraction must be deuterated. In neutron scattering experiments the specific scattering length of atoms is used. The exchange of exchange of 1H by 2H (D) was introduced as the most important example, since the chemical modification is often negligible. But also natural substances might have a contrast with each other. In comparison, the scattering length density for x-ray experiments results from the electron density multiplied by the classical electron radius (2.82fm). So the scattering of x-rays is determined by the electrons. This means that the atoms are smeared out over the electron shell dimensions, which has to be taken into account for the data analysis. Especially, atomic structures are hardly resolved by x-rays. Furthermore, the scattering of x-rays is dominated by heavy atoms with larger Z-numbers. And isotope exchange does not lead to different contrasts. Especially for organic materials this is of disadvantage. Nonetheless, the high intensities at synchrotrons make experiments on organic materials with their low natural contrast feasible. So the density differences or slight differences in the chemical composition are sufficient for the desired contrast. 4. Absolute calibration In this section, the macroscopic scattering cross section is connected to the experimentally measured intensity. The experimental intensity is dependent on the instrument at hand, while the macroscopic scattering cross section describes the sample properties independent of instrumental details. The absolute calibration allows to compare experimental data between different measurements. In theory, the intensity and the cross section are connected by:

(7) The intensity ΔI for one detector channel is measured as a function of the scattering angle. Each detector channel covers the solid angle ΔΩ. The experimental intensity is proportional to: (a) the intensity at the sample position I0 (in units of neutrons per second per area), (b) of

8

the illuminated area A, (c) the transmission of the sample (the relative portion of non-scattered neutrons) , (d) of the sample thickness d, and (e) the macroscopic scattering cross section dΣ/dΩ. In most practical cases, the primary intensity cannot be detected by the same detector. By a calibration measurement of a substance with known scattering strength the primary intensity is measured indirectly. At KWS-1 and KWS-2 we often use plexiglass, which scatters only incoherently (due to the hydrogen content). The two measurements under the same conditions will be put in relation, which thereby eliminates the identical terms. One writes:

(8)

The macroscopic scattering cross section of the plexiglass measurement does not depend on the scattering vector. The measured intensity of the plexiglass is also a measure of the detector efficiency, as different channels can have different efficiency. The plexiglass specific terms are merged to . So, finally the macroscopic

scattering cross-section reads:

(9)

Essentially, formula 9 follows directly from equation 8. The last factor results from the solid angles of the two measurements, which in principle can be done at different detector distances D. Plexiglass is an incoherent scatterer, and therefore can be measured at smaller detector distances to obtain an increased intensity. Nonetheless, the collimation setting must be the same as for the sample measurement. 5. Form factors We refer to chapter 3, and especially to formula 5, and consider spherical domains or colloids in a solvent. The integral will be taken over the spherical volume, with the scattering length density being constant (homogenous). For spherical coordinates we obtain:

(10) One assumes, that the coordinate system of the sample is based on the scattering vector Q, such that only the Z component remains in the scalar product. The integral over the polar angle results in a factor of 2π. The azimuth angle can be integrated easily: (11)

This immediately yields:

9

(12)

The following factors become visible: The scattering length density, the volume of the sphere, and a fraction which describes the shape of the sphere. This last fraction is referred to as the form factor in many references. Throughout this manuscript we will call the square of this term the form factor. But first, we consider the sphere in a solvent. The scattering length densities we call now ρsphere and ρsolvent. For the amplitude of the whole space we obtain:

(13)

The space of the solvent is the sum of the entire space minus the spherical volume. This difference results in two terms with the prefactor ρsolvent. For the Fourier transformation of the whole space we obtain the usual Dirac delta function, and the Fourier transform of the sphere we have already calculated. As summarized in equation 13 the term for the sphere bears a prefactor from the scattering length density difference, which is also called the contrast. This part of the function we will also observe in the scattering experiment. The second term, i.e. the Dirac delta function, is invisible in the experiment, due to its sharpness. Practically, this term has a finite sharpness given by the sample size. The sample size is in the cm region, and is not resolved in a small angle scattering experiment. Thus, we neglect the Dirac delta function, and calculate the macroscopic cross section to:

(14)

The macroscopic scattering cross section contains the following important factors: (a) the contrast (Δρ)2, resulting from the scattering length density difference, (b) the concentration, obtained from the volume of the colloid with respect to the total volume (for more colloids the concentration still describes the sum of all scattering contributions), (c) the volume of a single colloidal particle and (d) the form factor describing the shape of the particle. The principal factors of equation 14 will maintain for other independent (non-interacting) colloidal particles of any shape. The special form factor F(Q) for a sphere is shown in Figure 4. For small scattering vectors Q the limit of the form factor approaches 1. In this region it is valid to develop the form factor as a Taylor series and we obtain: F (Q) = 1 - (QR)2/5. Thus the decay of the form factor at small Q describes the particle size. This behavior is also described by the Guinier approximation, but considers (F(Q)) as a function of Q2. The Q range of the Guinier approximation describes length scales, observing the whole particle. Also the first minimum describes the particle size, Qmin = 4.493/R. This formula illustrates again the relationship between the real and the reciprocal space. Large particles have their minimum at small Q, and small particles their minimum at large Q. For very large scattering vectors strong oscillations arise. Averaged over the oscillations, a decrease in intensity remains with a scattering law dΣ/dΩ = P∙Q-4, Porod’s law. The Porod constant is P=2π(Δρ)2Stot/Vtot. It is a measure of the total area in the entire volume. For spheres this can also be expressed by their radius, which yields P = 6π(Δρ)2фK/R. The original interpretation as

10

a measure of area per volume is however a generalization that applies to all sharp surfaces with random orientation. For observing domains, which forms are not well known, Porod’s law makes it at least possible to gain some information about the surface. Diffuse surfaces result in an even greater decay in intensity, which may extend to Q-6. Now a formula for the macroscopic scattering cross section of an infinitely long cylinder with a finite radius will be discussed. This cylinder is assumed to be infinitely long, because the experimental Q-window is not sufficient to observe the entire particle as a whole.

Figure 4: Form factor of a homogeneous sphere of radius R. For small scattering vectors Q the form factor approaches a value of 1. The initial decay at small Q reveals the size of the particle, as does the first minimum. So our scattering formula does not have a Guinier region for the entire cylinder. Moreover, the question about the orientation of the cylinder arises, because this particle is no longer isotropic. Simple sample preparations (like in the case of the lab course) do not impose a preferred orientation on the particles and they are oriented arbitrarily. The presented macroscopic scattering cross section was calculated with an orientation-average, which however will not be discussed in detail. One finds:

(15) Again, you get individual factors with the following meanings: (a) contrast, (b) the concentration of the cylinders, (c) the cross-sectional area of the cylinder, (d) a form factor part for infinitely long cylinders and (e) a form factor part of the shape of the cross-sectional area, which is simply a circle. Since the Guinier region of the entire cylinder slides out of the

0.1 1 1010

-5

10-4

10-3

10-2

10-1

100

F(Q

R)

QR

11

Q-window, the observable volume (CV from Eq. 14) is reduced to a cross section AZ of the cylinder. In addition, the total form factor can be expressed as the product of two terms. This is possible if the entire structure can be expressed as a convolution of two simpler structures. In this case, this is an infinitely thin infinitely long cylinder and a flat disk perpendicular thereto. The convolution is a cylinder with a finite cross section. The infinitely long cylinder is described by the term π/Q. While having strong oscillations for finite lengths, in the limiting case this factor yields just a power law - similar to the Porod-law. The orientation-averaged disc has a rather complicated expression with a first-order Bessel function. The first zero is at Q = 3.832/R. This factor can also be regarded as a separate form factor with its own Guinier region. In this case, Eq. 15 would be written as follows:

(16)

These two principal areas of Guinier regions derive from a scale separation. On the one hand, the cylinder is very long, and the first Guinier region disappears at low Q. On the other hand, one finds a Guinier region of a circular disk of radius R in our Q-window. Both Guinier regions are clearly separated in reciprocal space. It is just that the observed Guinier region is covered by an additional form factor for an infinitely long, infinitely thin cylinder. The radius of the cylinder can be determined from scattering data, when plotting ln(I•Q) as a function of Q2, with I being the intensity. 6. Structure factors This section is supposed to describe the influence of finite interactions. Most of these are visible in the scattering pattern if the concentration is sufficiently large so the particles encounter each other often. On average, one observes an additional factor that arises from the interactions. Repulsive interactions suppress the intensity at small scattering angles, while attractive interactions increase this intensity. The suppressed intensity can be interpreted as an order, while the increased intensity can be interpreted as a temporary or transient cluster. This manuscript will only be talking about repulsive interactions. Now a structure factor in its simplest form will be developed. We therefore consider all the particles in pairs, and in this sense, first of, just two particles. Each particle is a spherical colloid. We again start with equation 5, which describes the amplitude. But now two particles are to be considered with different centers R1 and R2. One writes:

(17)

with

The contrast was already used, and thus the δ(Q)-contribution of the whole volume has been neglected. Important for two particles are now the phases resulting from the suspension points R1 and R2. On the one hand, these are just two shifts towards the origin. On the other hand one can interpret the form of Eq. 17 as a convolution of two points with a

12

sphere. The shape of the sphere is isotropic in any case, and thus does not dependent on the direction of the Q-vector. (Note: from my old German manuscript I kept the notation K(Q) and VK for the sphere = Kugel). For the macroscopic scattering cross section (see Eq. 4) we get:

(18)

The phase of the first particle is squared, and is therefore 1. The terms of the relative phase will now be statistically averaged. We consider the simple case where the spheres are not able to penetrate each other, but otherwise there is no potential. The last term is the form factor of a sphere, as we know from Eq. 14. In the following we will focus on the statistics of relative particle centers. The following applies:

(19)

(20)

The last step was integrated over all possible ΔR. The constant term is thus the entire volume. The cos-term is again a Fourier transformation, which leaves out a volume of radius 2R. Consequently, we obtain a δ-function and a contribution from the left-out spherical volume. All these contributions are normalized to the total volume. The resulting structure factor now looks like this:

(21)

and this is:

(22)

The prefactor 2 in Eq. 20 is now included in the average particle concentration фK. Equations 21 and 22 describe, the scattering of spherical colloids with the interaction due to excluded volume. However, this approximation was only derived for dilute solutions, which only considers pair-wise interaction. A simplified Ornstein-Zernicke approximation would, instead of Eq. 21, yield the following: S(Q) = 1/(1+фK(2R)K(Q,2R)). One can, however, solve the interaction of the excluded volume exactly, and receive the somewhat more complicated Percus-Yevick structure factor [1]. The idea behind the structure factor becomes clear however, even in this simple approximation for dilute solutions. Graphically, an example of a structure factor is shown in Fig. 5. The following structure factor properties are important: For small Q the structure factor is less than 1. This is found for repulsive interactions. The reduced scattering at small Q means that the sample appears more ordered on long scales as an ideal solution. The structure factor has a first maximum at about QR = 0.3. So the particles primarily assume one specific distance. Would the interactions be more repulsive or the concentration much higher, the structure factor at small Q would disappear entirely, and the Maxima would become Bragg peaks.

13

0.1 1 1010

-5

10-4

10-3

10-2

10-1

100

0.8

1.0

F(Q

R)

QR

S(Q

R)

Figure 5: Structure factor and form factor of a spherical colloid solution. The structure factor was calculated for a rather high concentration of 30% in order to make the oscillations visible. Here, the approximation of this script would be no longer valid. The form factor corresponds exactly to Fig. 4. Note that here F(QR) = K2(Q,R) from the text. In the lab course charged proteins are examined. Their repulsive interaction is already so strong that the scattering shows a significant peak at a Qmax, while the scattering is strongly suppressed at small Q. Such structural factors are usually calculated numerically. A qualitative understanding has already been achieved with Eq. 21. Instructions for the experiments 1. The protein lysozyme in water For this part of the experiment a Lysozyme solution of 0.02g per ml of water must be prepared. We will use deuterium to prepare the solution of ca. 1ml. We will weigh exactly 0.02g of Lysozyme and put it into a new Packard glas. With an Eppendorf pipette we will add exactly 1 ml D2O. These pipettes are extremely accurate with respect to the volume. From the solution about 0.5 to 0.6ml are transferred to Hellma quartz cuvettes, which are 1 mm thick. The experimental part in the chemistry lab is finished. 2. The diblock copolymer POO10-PEO10 This polymer is an amphiphilic diblock copolymer. It consists of the monomers octyl-oxide and ethylene oxide. In the polymer these monomers were successively polymerized. The

14

POO-block has a molar mass of 10kg/mol. The symmetrical polymer has about the same molecular weight in the PEO block. The PEO is deuterated in this polymer. Using heavy water especially the contrast of the POO-hydrophobic block to PEO and D2O is highlighted. This means that in the scattering experiment the micelle core will be particularly well visible, the corona will be not. Because these polymers are very expensive, a 0.1% solution is available for the scattering experiment. 3. The measurement at KWS-1 (KWS-2) These seven solutions are now being measured in the small-angle neutron scattering instrument KWS-1. The wavelength of neutrons is set to 7Å. The collimation is fixed to 8m. The samples are placed as close as possible to the detector, to measure the largest Q values possible. Both samples will be measured at detector distances 2m and 8m. The offset of the sample position of about 30cm leads to effective detector distances of about 1.7m and 7.7m. The sample holder will be filled with the two samples. In addition, the empty beam and a plexiglass plate are measured for absolute calibration. For a good statistical measurement the following times are set: 8m detector distance for 20min, and 2m detector distance 10min. The total measuring time for the 4 positions will be about 2 hours. The measurement is typically started before lunch, and can be evaluated in the afternoon. It is quite likely that an internal employee will start separate measurements during the afternoon until the next morning in order to use the valuable measuring time overnight. 4. Evaluation of the measured scattering data: absolute calibration The measured data is raw data at first and describes the intensity on the detector. The data has to be corrected for the effectiveness of the different detector channels. Then the empty beam measurement is deducted to account for the zero effect of the instrument. Then the intensities are expressed as absolute units using Eq. 9 and are radially averaged, because for the isotropic scattering samples, the intensity does not depend on the polar angle. To perform all these steps we will be using a software available in our institute, called QtiKWS. However, since the understanding of the Eq. 9, as such, is more important than the exact technical understanding of the evaluation, the results are produced relatively quickly by the software, namely, dΣ/dΩ as a function of the scattering vector Q for our samples. This data will be provided for the students to do the final evaluation. In the following, this evaluation is described. 5. Evaluation of Lysozyme scattering curves The position of the maximum Qmax provides information on the typical distance of the proteins in solution. This can be calculated to ℓ = 2π/Qmax. Knowing the weight of the protein in water (0.1g/5cm3) there is an alternative way to calculate the average distance. The molar mass of the protein is 1.43∙104g/mol. The number density of the protein is therefore n/V=0.1g/5cm3/1.43(g/mol)=1.40∙10-6mol/cm3=8.42∙10-7Å -3. For a simple cubic packing the typical distance is given by ℓ = √3(V/n). For a hexagonal packing the typical distance is ℓ = 21/6 √3 (V/n). Both calculated distances are to be compared with the measured one.

15

6. Evaluation of POO10-PEO10-scattering curves The entire scattering curve is first plotted in a log-log representation (Fig. 9). While in the shown diagram, the curve is measured to even smaller Q, a larger Q values suffice for the lab course ranging from about 0.006 up to 0.2Å -1. We see a strong drop in intensity at about 0.01Å -1. In this area the Guinier region of the cylinder cross section is found. Note that the contrast was chosen in a way, such that only the micelle core is visible. At higher Q two minima and maxima are found, which describe the sharp boundary of the micelle core. These two oscillations become clearly visible when the incoherent background is subtracted. It will be sufficient for the lab course to calculate the average of seven data points from the highest scattering vectors and subtract it. The quality of the measurement is only visible in this representation. The basis for our further analysis is the Guinier-plot (Fig. 10). In the Gunier-plot ln(Q∙dΣ/dΩ) is plotted against Q2. On the one hand, the data points show no significant slope for the smallest Q. On the other hand the examined Q values cannot be too big, because then the Guinier analysis is no longer valid. From the slope of the regression one can extract the diameter of the cylinder cross section (see Eq. 16). This is in the range of 120Å. For the validity of the Guinier assumption, the product of the highest Q value taken and the obtained radius can be calculated. This yields 0.02Å -1∙120Å = 2.4. This value is not much larger than √4. Therefore, the Guinier approximation is a good estimate for the cylinder diameter. The measurement error of the obtained radius is to be specified.

10-3

10-2

10-1

10-4

10-3

10-2

10-1

100

101

102

Daten x 0.1,

Inkoh鋜enter

Untergrund

abgezogen

D = 8m, = 19ÅD = 8m, = 7Å

D = 2m, = 7Å

d

/d

[cm

-1]

Q [Å-1]

Figure 9: Measured scattering curve of POO10-PEO10 micelles. The bright green data are measured at a longer wavelength, and are not intended to be repeated in the lab course. Here you can see, however, the Q-1 behavior, which clearly speaks for the cylindrical shape of the micelle core. The dark green and blue dots are measured in the lab course. The sharp drop of the curve at about 0.1Å -1 shows the Guinier region of the cylinder cross section. This is evaluated in further detail. Two minima and maxima follow at higher scattering vectors Q before the incoherent background is noticeable. Black data points, shows the curve without background (moved down to clarify the distinction).

16

0 1x10-4

2x10-4

3x10-4

4x10-4

-2.4

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

ln(Q

d

/d

[Å

-1cm

-1])

Q2 [Å

-2]

Figure 10: Guinier plot for cylindrical cross section. The first data points of the 8m-7Å measurement indicate no slope. The data points for larger Q values than 0.02Å -1 are neglected. Alternatively, the first minimum Qmin in Fig.9 can be used for calculating the cylinder cross section. The radius is calculated according to R = 3.832/Qmin. The two different radii are to be compared. Is the Guinier condition QR < 2 meet in good accordance?

Literature:

[1] X. Ye et al., Phys. Rev. E 54, 6500 (1996) siehe speziell Gl. A5

17

Excercises for students in the morning

(I) Lysozyme in D2O.

The first sample of the Neutron Lab Course at the SANS instrument KWS-1 will be Lysozyme in heavy water (D2O). This protein is rather globular (diameter ca. 5 nm). The Coulomb interactions of this charged molecule lead to liquid-like short-range-ordering. This will be observed in the SANS scattering experiment by a correlation peak. Simple estimations will be made now:

1.) Give the connection between the number density φ and the unit cell parameter

assuming a simple cubic lattice!

2.) The chemical concentration c is usually given in g/L or mg/ml. The molar mass of the

molecule is 14,307 g/mol. What is the connection between the chemical

concentration and the number density?

3.) The correlation peak appears at a scattering vector Q. How would it relate to the unit

cell parameter of a simple cubic lattice? What is the dependence of Q as a function of

the chemical concentration c?

4.) If the packing of the globules was fcc (face centered cubic) or hexagonal (with

arbitrary layer order) the system is more dense by a factor of 21/6. What would be the

function Q(c)?

(II) Cylindrical micelles in D2O.

The Q-dependence of the scatting curve will be evaluated in 2 different ways to obtain the

diameter of the cylinders.

1.) The Taylor expansion for small scattering vectors Q yields the following functional

form:

)exp()( 22

411 QRQQI

What would you plot in a simple graph to obtain a linear dependence? Which role

does the radius R play in this graph?

2.) Knowing that the formula from II-1 occurs from a Taylor expansion with the

parameter x = QR around the point x = 0, what would be criterion for a good

approximation? (Don’t argue too mathematically!)

3.) As for many compact (and monodisperse) bodies the scattering curve shows

oscillations. The first minimum of such a curve also indicates the radius of the

cylindrical micelle. Mathematical calculations yield:

RQ /832.3

This is another example for connections in real space to reciprocal space. Where does

the name reciprocal space come from?

4.) The mathematical form of an ideal, infinitely long cylinder would predict a zero

intensity for the Q-value given in II-3. Why would we observe a finite intensity

nonetheless?