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X-ray Photon Correlation Spectroscopy Studies of Surfaces and Thin Films Sunil K. Sinha,* Zhang Jiang, and Laurence B. Lurio
changing phase differences between the scattered waves. It was recognized about 50 years ago that this dynamical fluctuation of brightness on a given area of the detector could yield information about the dynamics of the object. Thus with the advent of highly coherent laser beams, this method, i.e., Dynamical Light Scattering (DLS), or Photon Correlation Spectroscopy (PCS), became well established[2] as a method for studying the diffusion of colloids or entities such as proteins in solution. One of the main applications of this technique was to obtain a quick method of obtaining particle sizes, since the diffusion coefficient for a dilute solution of colloids, etc. is related to the hydrodynamic radius of the particle, the viscosity of the solution and the temperature via the StokesEinstein relation.[3] DLS yields dynamical information at fairly small values of the wave vector transfer q (i.e. rather large length scales), being limited to dimensions greater than the wavelength of visible light. While it was recognized that carrying out similar experiments with X-rays would make it possible to study larger values of q (i.e. shorter length-scales), such experiments were not possible due to the relatively low brilliance of available X-ray sources, until the advent of high-brilliance synchrotron X-ray sources, beginning with the so-called third generation light sources. The high brilliance of these and subsequent sources have made it possible to obtain a sufficiently high flux of coherent photons from such sources (in a manner to be discussed below) to be able to carry out photon correlation spectroscopy studies using coherent (or partially coherent) beams of X-ray photons, thus giving rise to the technique of X-ray Photon Correlation Spectroscopy (XPCS). The technique of XPCS is accessible at a number of coherence beamlines at synchrotron facilities worldwide, among which are a few dedicated ones such as Sector 8-ID-I at the APS (Advanced Photon Source, USA), ID10A Troika I at the ESRF (European Synchrotron Radiation Facility), P10 at PETRA III, DESY (Germany), as well as the newly constructed CHX at the NSLS II (National Synchrotron Light Source II, USA). Within little more than a decade, experiments using this technique have gone from being simple demonstrations of a novel technique to actually being used to obtain information difficult to obtain by other means. XPCS has had a particularly significant impact on soft condensed matter research, such as the study of the dynamical
The technique of X-ray Photon Correlation Spectroscopy (XPCS) is reviewed as a method for studying the relatively slow dynamics of materials on time scales ranging from microseconds to thousands of seconds and length scales ranging from microns down to nanometers. We focus on the application of this technique to study dynamical fluctuations of surfaces, interfaces and thin films. We first discuss instrumental issues such as the effects of partial coherence (or alternatively finite instrumental resolution) and optimization of signal-to-noise ratios in the experiments. We then review what has been learned from recent XPCS studies of capillary wave fluctuations on liquid surfaces and polymer films, of nanoparticles used as probes to study the interior dynamics of polymer films, of liquid crystals and multilamellar surfactant films, and of metal surfaces, and magnetic domain wall fluctuations in antiferromagnets. We then discuss studies of non-equilibrium dynamics described by 2-time correlation functions. Finally, we briefly speculate on possible future XPCS experiments at new synchrotron sources currently under development including studies of dynamics on time scales down to femtoseconds.
1. Introduction When a coherent beam of light falls on an object with any type of disorder (static or dynamic), the scattered light is broken up into bright and dark regions as seen in a 2D detector or on a screen. These are due to the irregular phase differences and hence irregular interference between the waves scattered from different parts of the object, and this apparently random array of bright spots is termed a “speckle pattern”, a phenomenon that has been known for years.[1] If different parts of the object fluctuate with time, the speckle pattern also fluctuates due to the Prof. S. K. Sinha Dept. of Physics University of California San Diego 9500 Gilman Drive, La Jolla, CA 92093–0319, USA E-mail: [email protected] Dr. Z. Jiang Advanced Photon Source Argonne National Laboratory Argonne, IL 60439, USA Prof. L. B. Lurio Dept. of Physics Northern Illinois University DeKalb, IL 60115, USA
DOI: 10.1002/adma.201401094
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Dr. Sunil K. Sinha is Distinguished Professor of Physics in the Department of Physics at the University of California, San Diego. He received his Ph.D. in Physics from Cambridge University. He has served as Group Leader in Neutron Scattering at Argonne National Laboratory, Group Leader of X-ray Scattering at Brookhaven National Laboratory, Senior Research Associate at Exxon Corporate Research Laboratories, and Associate Division Director at Argonne’s Advanced Photon Source. His group’s research is concerned with studying the structure and dynamics of condensed matter using the techniques of x-ray and neutron scattering.
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behavior of colloids,[4] liquids,[5] polymers[6] and clays.[7] This is not surprising since, in some sense, what makes soft matter “soft” is its ability to exhibit significant thermal fluctuations near room temperature. However, the impact of XPCS has not been limited to this area, and has also been used in areas such as the study of magnetic domain fluctuations,[8] phase ordering in glasses[9] and metal alloys,[10] atomic diffusive motion in crystals,[11] martensitic phase transformations[12] and the ordering of single crystal interfaces.[13] There already exist several excellent reviews dealing with XPCS in the literature.[14] In this article we will briefly describe the technique and related instrumentation, and discuss recent applications of this technique to surfaces, interfaces and thin films. The recent development of new sources with a much higher degree of coherence, such as the X-ray free electron laser sources (XFEL’s) or the planned diffraction-limited storage ring (DLSR) sources will make it possible to greatly extend both the timescales and length scales which can be studied with XPCS, and we will briefly discuss prospects for future experiments with such sources later in this article. Techniques such as DLS, XPCS and neutron spin echo probe the sample scattering function in real time, and as such are complementary to inelastic scattering (of light, X-rays or neutrons) which probe the scattering function in frequency space. For slow dynamics the former are more convenient as the energy resolution required for inelastic scattering becomes too stringent. Interestingly, the results of DLS experiments are often Fourier transformed (by using the power spectrum of the scattered light, e.g. using a spectrum analyzer) to give the scattering function as a function of frequency.[15] The timescales which can be studied with XPCS are primarily limited by the flux of coherent photons falling on the sample, since the ability to resolve short-time dynamics relies on sufficient flux being scattered into a speckle within the time-scale of interest. A figure of merit that has been used in dynamic light scattering is that an experiment requires at least one scattered photon per speckle over the timescale of interest. In fact, this metric is too conservative, since it has been shown that with the use of area detectors many speckles can be measured at once, with a concomitant increase in signal.[16] However, if an area detector is used, it must be able to read out within the relevant timescale and the available readout speed of existing x-ray area detectors then becomes the limiting factor in many cases. Current XPCS experiments have studied dynamics on time scales from microseconds to thousands of seconds. In this sense they are complementary to neutron spin echo experiments which probe dynamics of systems in real time on time scales of nanoseconds and shorter. The wavevector transfers (q values) which can currently be studied by XPCS can range from 10−3 nm−1 to several nm−1. (See Figure 1). The transverse coherence length of a photon beam can be interpreted in terms of the familiar Young’s double slit experiment (see Figure 2(a)) in terms of how far apart the two slits can be before the visibility of the fringes due to the photon beam on the far screen drops below some given criterion, i.e the characteristic distance in the direction transverse to the beam over which there is phase coherence of the beam. There are evidently two such transverse coherence lengths and they are given by the formulae
Dr. Zhang Jiang is currently a scientist at the Advanced Photon Source, Argonne National Laboratory. He received a B.S. in applied physics from University of Science and Technology of China in 2001 and a Ph.D. in physics from University of California, San Diego in 2007. Since then he has been working at the Advanced Photon Source, where he is responsible for the operation, development and maintenance of the X-ray photon correlation spectroscopy and grazing-incidence X-ray scattering beamlines. His scientific research mainly focuses on using coherent X-ray scattering and grazing-incidence surface scattering to study the structure, kinetics and dynamics of liquids, soft matter, and nanocomposite materials. Dr. Laurence Lurio is a professor at Northern Illinois University and chair of the physics department there. He received a B.A. in physics from Columbia University and a Ph.D. in physics from Harvard University. He was part of the team which constructed and commissioned the dedicated XPCS beamline at sector 8-ID of the APS, and is currently a member of the beamline access team planning the first experiments at the coherent hard x-ray (CHX) beamline at NSLS-II. He has also served as chair of the APS User Organization Steering Committee. His research interests focus on structural and dynamical properties of condensed matter with a particular emphasis on liquids, polymers and biomaterials.
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pattern, while the speckle intensity will diminish as the pinhole size is increased beyond that size. There is also a limitation on the longitudinal coherence, which is governed by the monochromaticity of the beam. The longitudinal coherence can be related to the maximum difference in the length of the arms of a Michelson interferometer which can be allowed and still obtain an interference pattern. It is given by ⎛ λ ⎞ ξl = λ ⎜ ⎝ Δλ ⎟⎠
Figure 1. Illustration of regions of frequency-wave vector space covered by various dynamical probes, including Inelastic Neutron Scattering (INS), Inelastic X-Ray Scattering (IXS), Neutron Spin Echo, Nuclear Forward Scattering (NFS), Photon Correlation Spectroscopy, i.e., Dynamical Light Scattering (PCS) and X-ray Photon Correlation Spectroscopy (XPCS). Reproduced with permission.[14d] Copyright 2008, Springer.
ξx =
λR λR and ξ y = 2πσ x 2πσ y
(1)
where λ is the average photon wavelength and R is the distance to the source. Here the source is assumed to have an incoherent Gaussian intensity profile of the form I( x , y ) ∝ exp ( − x 2/2σ x2 − y 2 /2σ y2 ). A beam passing through a pinhole of dimensions ξxξy will yield a well-defined speckle
Figure 2. (a) Interference pattern arising from a Young’s double slit experiment with a finite source size. Rays originating from the top and bottom of the source produce offset interference patterns which limits the maximum transverse separation of the slits which can still yield an interference pattern. (Colors are for clarity and not meant to imply differing frequencies). (b) The longitudinal coherence length ξl can be related to the width of the wave packet made up of a small distribution of wavelengths over which interference is possible.
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(2)
where Δλ is the wavelength spread of the beam (see Figure 2(b)). This is typically defined as the full width at half maximum of the wavelength spread of the beam, but the precise longitudinal coherence length will depend on the functional form of the wavelength spread. The transverse coherence lengths of the radiation actually incident on the sample will also be affected by X-ray optics, slits, etc. in the beam path prior to the sample. The product of these three lengths is proportional to the coherence volume of the beam, i.e. the volume within which the scattering is coherent. The brightness B of a source is defined as the number of photons emitted per second per unit source area per unit solid angle. Thus the number of transversely coherent photons passing per unit time through a pinhole of dimensions ξxξy is proportional to I coh =
π 2Bσ x σ y ξx ξy λ 2B = R2 4
(3)
However, if the source is not highly monochromatic (e.g. a particular harmonic of an undulator at a 3rd generation source), then ξl will be quite small. To increase it, one needs to monochromate the beam further, for example using a crystal monochromator (where Δλ/λ can be ∼10−4), so that the number of coherent photons will then be proportional to the spectral brightness (sometimes referred to as brilliance) of the source, which is defined as the brightness per 0.1% bandwidth in energy. For current third generation undulator sources, the transverse coherence lengths are of the order of 5 μm to 200 μm in the vertical direction and 2 μm to 5 μm in the horizontal direction while the longitudinal coherence length can be 1–2 μm after monochromatization. The effects of partial coherence will be discussed in more detail below. In general, the scattering from 2 points separated by a distance Δr can interfere coherently if Δr is less than both (not “either” as stated in Ref. [17]) ξt /sin α i and k0 ξ1 /q where αi is the angle the incident beam makes with Δr,k0 is the magnitude of the incident wavevector and q is the component of the wavevector transfer parallel to Δr[17] (see Figure 3). Thus we see that the effective transverse coherence length in the plane of incidence is greatly magnified if αi is small (e.g. if we have scattering from a surface under conditions of grazing incidence) and the effective longitudinal coherence length is magnified for small q values, or for scattering from a surface in directions close to specular. If the coherence volume is much greater than the sample volume, then all points in the sample scatter coherently, and
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Figure 3. Illustration of the maximum separation of two points on a surface in order to produce coherent scattering. Transverse coherence requires Δr < ξt /sinα i , while longitudinal coherence puts a maximal limit on the X-ray path difference (bold line segments), i.e., Δr < k0ξl /q|| with q|| = k0 ( cos α i − cos α f ) the in-plane component of the wavevector transfer.
the speckle pattern is determined by the exact positions of all scatterers in the sample, i.e. it represents a “fingerprint” of the sample. If the beam is only partially coherent, the visibility of the speckle pattern decreases. A recent discussion of the statistical properties of the intensities of speckles and their visibility has been given by Grubel, Madsen and Robert,[14d] who show that the speckle contrast goes to zero as the coherence volume decreases relative to the scattering volume. The dynamics of a system is usually characterized in terms of the dynamical structure factor or scattering function S(q,ω) which is given by S( q , ω ) =
1 2π
∞
∫ dte
− iω t
ρ( q ,0)ρ( − q , t )
(4)
−∞
where ω is the energy loss of the probe, and ρ(q,t) is the Fourier component of the electron density at time t. The average is either a statistical or quantum mechanical average over the system. This is the function which is measured in conventional inelastic scattering experiments. The frequency Fourier transform of S(q,ω) is the function g1(q,t), also known as the intermediate scattering function. This is given by g 1 ( q , t) =
1 ρ( q ,0)ρ( − q , t ) N
(6)
where C is a constant related to parameters such as the incident beam intensity, etc. and the average is evaluated at time t. Intensity correlation experiments are of two categories: (a) homodyne experiments where the time variation of the scattered beam alone is measured and (b) heterodyne experiments, where the scattered beam is made to interfere with a static reference signal. Most XPCS experiments are carried out in the homodyne mode, although this is by no means true of DLS.[18] What is measured in a homodyne XPCS experiment is the intensity autocorrelation function given by
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(7)
where the average is over the time t′. Note that as written g2(q,t) is fourth order in the electric fields, since the intensities are each second order in the fields. One can often write the scattered electric fields (which are proportional to ρ(q,t) as Gaussian random variables with zero mean. Then the above equation can be decomposed into a simpler product of correlation functions of electric fields, rather than intensities i.e. g 2 ( q , t) = 1 + F( q , t)
2
(8)
where F( q , t ) = g 1 ( q , t ) / g 1 ( q ,0) is the normalized intermediate scattering function. This is known as the Siegert Relation.[2] Note that in this case the phase of g1(q,t) is undetermined. If the beam is only partially coherent, so that the speckle contrast is low, and the length scales associated with the sample structure are significantly smaller than the illuminated sample dimensions, then this relation is often modified[19] to read g 2 ( q , t) = 1 + β ( q ) F( q , t)
2
(9)
where the q-dependent contrast factor β(q) depends on the incident beam properties and the experimental setup. It can in principle vary from 0 to 1. We postpone discussion of this quantity and the discussion of the effects of partial coherence and resolution effects to Section 2. The dynamics of the system in terms of the intermediate scattering function can be inferred from g2(q,t) –1. A heterodyne experiment in principle can be used to deter2 mine both the real part of g1(q,t) and g 1 ( q , t ) . For details see Refs. [18,20] The optimization of experiments at 3rd generation sources is often governed by the need to balance the requirement for a high counting rate to obtain better statistics for the function g 2 ( q , t ) in a given time, and the requirement for better contrast, which often requires lower absolute beam intensity.
(5)
where N is the total number of scatterers. For a completely coherent X-ray beam, the scattering intensity from the sample for a given q (no energy analysis) at any instant of time t is given by I ( q , t ) = C ρ( q , t ) ρ( − q , t )
2
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g 2 ( q , t ) = I( q , t ′)I( q , t ′ + t ) / I( q )
2. Methods We now discuss briefly some of the general types of studies carried out with XPCS. (a) Transmission XPCS experiments and colloidal diffusion While such experiments are not the main subject of the present review, we mention them in passing for historical reasons, as the earliest XPCS experiments were carried out on colloidal particles in solution.[4] These experiments are carried out in transmission geometry as in small angle X-ray scattering experiments. What is probed is the dynamics in the bulk of the sample, such as the diffusion of particles in solution. For a system of discrete particles the Fourier component of the electron density is given by ρ( q , t ) = ∑ f i (q)e − iq . r ( t ). Here f i is the form factor associated with particlei i . In the simplest case where all particles have the same size one obtains:
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g 1 ( q , t) =
f 2(q) N
2
∑e
− iq . ri (0) iq . rj ( t )
e
(10)
i, j
In a system where the particles are undergoing random Brownian motion and where their motion is uncorrelated, the averages for i ≠ j and finite q vanish and we are left with g 1 ( q , t) =
f 2(q) ∑ e − iq .[ri (0)− ri ( t )] N i
(11)
Using the fact that the ri are Gaussian random variables, we can 2 1 − {q .[ ri (0) − ri ( t )]} f 2(q) e 2 write g 1 ( q , t ) = and using the diffusion ∑ N i 2 law (ri (0) − rj (t )) = 6Dt and averaging over all orientations of q and [ ri (0) − ri (t )] , we obtain the well-known result for hydrodynamic diffusion 2
F( q , t ) = e − Dq t
(12)
Thus for hydrodynamic diffusion, the relaxation time is 1/Dq2. As mentioned previously, this is often used in DLS to determine the diffusion constant of particles in solution, and hence, using the Stokes-Einstein relation, to obtain an estimate of the hydrodynamic radius of the particle. For interacting particles the form of the dynamic structure factor is generally more complex. However, one can attempt to maintain connection to this formalism by replacing D with a q-dependent function D(q,t).[19] (b) XPCS at Grazing Incidence and dynamics of liquid surfaces The most straightforward application of XPCS to the study of liquid surfaces is to carry out measurements using an X-ray beam which impinges on the surface at grazing incidence. For a free liquid-vapor interface what one typically measures in such cases is the dynamics of capillary wave fluctuations. The first such measurements were carried out by Seydel et al.[5a] on liquid glycerol and by Kim et al.[6b] on molten polymer films. There have also been many earlier DLS studies of capillary wave fluctuations on liquid surfaces.[15] For a liquid surface, the expression for the diffuse scattering cross-section at an instant of time t [21] yields I( q , t ) =
PA − q2zσ 2 e q z2
∫ ∫ dXdY {e
q 2z C ( R , t )
− 1}e − i [ qx X + qyY ]
(13)
Here P is a constant related to the incident beam intensity and the electron density of the liquid, A is the illuminated surface area, σ2 is the mean square roughness, R is the in-plane vector (x, y), and C(R, t) is the time dependent height-height correlation function C( R , t ) = u z ( r , t )u z ( r + R , t )
r
(14)
where u z ( r , t ) is the displacement of a point on the liquid surface at in-plane position r and time t. For small values of qz, the exponential in the integral can be expanded, leading to the approximate expression
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I( q , t ) = PAe − qzσ
2
∫ ∫ dX dYC(R , t)e
− i [ qx X + qy Y ]
(15)
The measured g2 function, using the Siegert relation, yields the corresponding g1(q,t) function: 2
2
g 1 ( q , t ) = P ′ e − qzσ u z ( q ,0)u z ( q , t )
(16)
where the correlation function is between in-plane Fourier components of uz(r, t). The vector q indicates the in-plane component of q. For liquids with low viscosity, such as water, we −Γ t have propagating capillary waves, so Re[ g 1 (q, t )] ~ cos (ω q t ) e q . The frequencies and damping constant are given approximately by[22]
ω = ( γ q ||3 / ρ )
1/2
and Γ = 2η q 2 /ρ .
(17)
Here γ is the surface tension, ρ the mass density, and η is the viscosity. For highly viscous liquids, such as polymer melts, the capillary waves are overdamped, so there is no cos (ω q t ) factor in Re [ g 1 (q, t )] . In this case homodyne XPCS experiments are sufficient to determine the damping constant and its q-dependence. Such experiments have been carried out for several molten polymer films, and will be discussed in more detail in the next section. The hydrodynamic theory of capillary fluctuations of a thin viscous liquid film on a solid substrate has been worked out by Jackle,[23] and by Harden, Pleiner and Pincus.[24] From their general formulation, Kim et al.[6b] derived the following expression for the relaxation time τ q (or Γ q−1 ) for overdamped capillary waves in a thin film of thickness h, surface tension γ and viscosity η:
τq =
2η[cosh 2 (qh ) + q2h 2 ] γ q [sinh(qh )cosh(qh ) − qh ]
(18)
From Equation (18), it may be seen that τ q / h is a function of only the dimensionless product qh and is proportional to the ratio η / γ . This scaling relation has been verified using grazing incidence XPCS for a number of polystyrene films of various thicknesses at several temperatures.[6b,25] If the liquid is viscoelastic rather than purely viscous, we shall see in Section 3(b) how Equation (18) above has to be modified to fit the data. Another advantage of grazing incidence scattering from thin films or surfaces is that by varying the grazing angle of incidence from below to above the critical angle for total reflection, and thus changing the penetration depth of the X-rays into the film, one can selectively probe the surface or underlying regions of the film. This method was employed by Hu et al. to study dynamics of buried interfaces in a polymer bilayer system.[26] (c) Effects of Partial Coherence and Resolution The measured intensity I( q ) in a detector in an actual experiment involves not just the intermediate scattering function g 1 ( q , t ) but also the partial coherence, characterized by the Mutual Coherence Function (MCF)[27] of the beam incident on the sample, as well as the finite detector acceptance angle. For beams which are not perfectly coherent and monochromatic,
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U α ( r , t ) = Aα ( r , t )exp( − iω t )
(19)
for the intensity in the detector (where we omit the time variable): 2
⎛ e2 ⎞ 1 I(q) = I 0 ⎜ 2 ⎟ 2 ∫ d Δq ⊥ P( Δq ⊥ ) ⎝ mc ⎠ λ
∫ ∫ drdr ′ ρ( r )ρ( r ′)e
iq ⋅(r − r ′ )
Γ( r⊥ , r ′ ⊥ , q ⋅ ( r − r ′)/ω )
where Δq ⊥ is the component of q perpendicular to the average scattered beam direction due to the finite size of the overall detector element and P( Δq ⊥ ) is the acceptance function of the detector (usually represented by a Gaussian function). Substituting for the MCF using Equations (21) and (22), we obtain:
where α is the polarization index, r denotes a point in space, and ω is the average frequency, while the amplitude represents a slowly varying (relative to the timescale of ω ) time-dependent function. The MCF is then defined by
⎛ e2 ⎞ 1 I(q) = I 0 ⎜ 2 ⎟ 2 ∫ d Δq ⊥ P (Δ q ⊥ ) ⎝ mc ⎠ λ
Γ α ( s, s′, τ ) = Aα ( s, t )Aα* ( s′, t + τ )
where
(20)
where s, s′ denote points in a plane perpendicular to the average beam propagation direction and the average represents a time average over many phase fluctuations of the radiation (which for X-rays have periods of 10−15 to 10−16 seconds).We shall henceforth assume completely plane polarized radiation and drop the polarization index α. In principle the MCF will contain the Fresnel diffraction effects from the slit prior to the sample, but for simplicity, we here assume a Gaussian-Schell Model[29] form for the MCF : Γ( s , s ′, τ ) = I 0 Ψ( s)Ψ * ( s′) g ( s − s′)F(τ )
(21)
where Ψ( s) represents an amplitude at s (which can be cut off if the beam cross-section is smaller than the sample dimensions) which is normalized to unity for maximum amplitude, I 0 is the incident beam intensity, and g(s − s′) and F(τ) can be represented by g (x , y) = e − x F(τ ) = e −τ /τ l
2
2 2 /2ξ x2 − y /2ξ y
e
(22)
where ξx and ξy are the transverse coherence lengths in the x, y directions respectively, and the coherence time τl is determined by the monochromaticity of the beam. Let us choose a plane in the sample perpendicular to the incident beam and choose the origin in the plane, so that the MCF at the sample position can be defined by Equations (21) and (22), and chose the direction of propagation of the incident beam as the z axis. We use the kinematic (HuygensFresnel) theory of wave propagation to describe the scattering as in Ref.[17] but here we derive a simplified version that describes the scattering in terms of the MCF at the sample instead of at the incident aperture, and neglect the higher order Fresnel terms (which is a good approximation if the correlation lengths of the scattering elements are small and the detector area is small –which is true if we are considering single pixels or small groups of pixels). This yields
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(23)
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this will result in both a loss of contrast (expressed in terms of the contrast factor β ( q ) in Equation (9) and smearing in q-space of the measured g 1 ( q , t ) function. Quantitative studies of the X-ray contrast and comparisons of experiments with theory have been made by several authors.[28] Those readers who are not interested in the details discussed here may skip to Section 2(d) below. The MCF is determined by the source and all the optical elements (including slits) prior to the sample and can be characterized in terms of the 2 transverse coherence lengths and the longitudinal coherence length. The complex analytical signal describing the incident electric field is given by
2
∫ ∫ d r d r ′ρ ( r ) ρ ( r ′ ) e
iq ⋅( r − r ′ )
R( r , r ′ )
(24)
R( r , r ′) = Ψ( r⊥ )Ψ * ( r⊥′ ) 2
2
e −( x − x ′ ) /2ξx e −( y − y ′ ) /2ξy e 2
2
− q ⋅( r − r ′ ) /k0 ξl
.
(25)
Here we have written ωτ 1 as k0 ξl to express it in terms of the longitudinal correlation length. R( r , r ′ ) acts like a resolution function which limits the coherent scattering from points separated by more than the transverse coherence length in the direction perpendicular to the incident beam and by more than k0 ξl in the direction along q . As seen from Equation (25), the longitudinal coherence length will not play a limiting role at small q for grazing incidence surface scattering, or if q is close to the specular direction. For small angle scattering q ⋅ ( r − r ′ ) is very small for r − r ′ along the incident beam, which implies the sample can be extended in this direction without loss of contrast. The longitudinal coherence length ξl has the property of limiting correlations contributing to scattering along the direction of q, resulting in streaking of the speckle patterns along q as we go to larger q. As an example, Figure 4(a) shows a speckle pattern from an aerogel sample measured using an X-ray beam monochromatized with a Ge-111 crystal with a bandwidth of ΔE /E = 3.3 × 10 −4 . The speckles appear to be completely isotropic. By contrast, the aerogel scattering pattern measured in Figure 4(b) was obtained using a pink beam (the direct undulator harmonic, without any monochromatization). The bandwidth for a pink beam is of order ΔE/E = 0.026. In this case the speckles are highly extended along the radial direction. A typical longitudinal coherence length ξl is, for example, ∼2 μm for a monochromatized beam of 8 keV with a bandwidth of ∼10−4. This limits experiments to very thin samples at high scattering angles. However, for small angle scattering, we can use samples as thick as several mm, before the effects of longitudinal coherence become significant. This has been noted by Sandy et al.[28c] Equation (24) yields the scattering from the exact electron density configuration at a given time, resulting in a speckle pattern, while the function R ( r , r ′ ) smears out this pattern in
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g 2 ( q ,τ ) = 1 +
2
⊥
P( Δq ⊥ ) ∫ d q ′ g 1 ( q ′, τ ) R 2 ( q − q ′)
∫ d q ′ g ( q ′,0)R( q − q ′)
2
(28)
1
Figure 4. Speckle patterns from (a) monochromatic incident X-ray beam, and (b) X-ray beam with broad energy bandwidth. Reproduced with permission.[28c] Copyright 1999, Intl. Union of Crystallography.
q-space, resulting in decreasing speckle visibility for smaller coherence lengths. It has been shown (see Refs. [14d,28a] that the speckle visibility is reduced by the number of coherence volumes contained in the illuminated sample volume. The integration over the finite solid angle of the detector (or group of pixels in the 2D detector) further smears the observed speckle pattern. For an actual measurement of I( q ) we average ρ( r )ρ( r ′) over time. For many systems which are ergodic and translationally homogeneous (such as fluids), this average is a function of (r-r′) alone, and if we assume constant values for the Ψ( r⊥ ) Ψ*(r′⊥) over the sample, then R ( r , r ′ ) can also be written asR ( r − r ′ ) . If its Fourier transform is R(q), then we have 2
⎛ e2 ⎞ 1 I( q ) = I 0 ⎜ 2 ⎟ 2 ∫ d Δq ⊥ P( Δq ⊥ ) ⎝ mc ⎠ λ
∫ dq
′
(26)
ρ( q ′ ) ρ * ( q ′ ) R( q − q ′ )
I( q , t )I( q , t + τ ) = I( q )
2
4
⎛ e2 ⎞ 1 + I 02 ⎜ 2 ⎟ 4 ⎝ mc ⎠ λ
∫ d Δq
2
⊥
P( Δq ⊥ ) ∫ d q ′ g 1 ( q ′, τ ) R 2 ( q − q ′)
(27)
(We assume here that the time correlations and the intensity normalization are performed for individual pixels and then averaged over the detector area). Using Equations (7), (26) and (27), we thus get the result for the effective g2 function including partial coherence and instrumental resolution effects as
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This expression is more complicated than the simplified expression (Equation (9)) commonly used, but shows the same behavior, namely a decrease of the second term on the right from the “ideal” value of |F(q,τ)|2 roughly in the ratio of the coherence volume to the total scattering volume. A discussion in terms of treating the speckles counted in the detector as “independent modes” by Lemieux and Durian[30] with speckle intensities I j (j = 1 to N) where N speckles are counted in the detector leads to a simpler result which has some similarities to Equation (28), namely g2 is given by Equation (9) with the parameter β given by N N 2 ⎛ ⎞ β = ∑ Ij /⎜∑ Ij ⎟ ⎝ j =1 ⎠ j =1
2
An interesting case arises when the function g1(q, τ) contains a static component (e.g. specular reflection in the case of surface XPCS, or a Bragg reflection) at a q-value close to what is being nominally measured, which contributes to the integrals in Equation (28). This then provides a heterodyne component to the measurement of g 2 ( q , τ ). This may change the form of g 2 ( q , τ ) from the true g 2 ( q , τ ). This has been discussed in detail in Ref. [87] for the case of propagating capillary waves, where by considering the effects of partial coherence, the specular reflection and detector acceptance (including the Fresnel factors neglected in the above derivation) it was shown that one can achieve a cross-over from cos2(ωτ) behavior to cos(ωτ) behavior in the measured correlation function, and the damping constants can be significantly different. For highly overdamped waves, the effect however is fairly small.[5b] (d) Focusing and Signal to Noise Ratios in XPCS
Equation (26) explicitly shows the smearing in q-space as a convolution of the true scattering function with an effective “resolution function”. For XPCS measurements, we are interested in the average I( q , t )I( q , t + τ ) over time t , as shown in Equation (7). Assuming the scattered electric field at the detector is a Gaussian random variable with zero mean, we can apply the usual decoupling scheme used to derive the Siegert relation, Equation (9). We omit the details of the calculation here, but the result is
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∫ d Δq
It is crucial in an XPCS experiment to optimize the signal to noise ratio due to the limited coherent flux available at current third generation synchrotrons as well as the need to reduce radiation damage to samples. A relation for the signal to noise ratio R sn in PCS in the limit of low intensity was given by Jakeman[31] and adapted to XPCS by Falus et al.[16b] This is given by: R sn = I β
TΔtN g2
(29)
Here I is the detector count rate (or count rate per pixel in the case of an area detector), β the contrast, Δt the integration time of the detected signal, T the duration of the experiment and N the number of independent detector channels averaged together to obtain the signal. A particularly interesting feature of R sn is that it depends linearly on the total incident intensity as compared with the usual expectation for incoherent X-ray scattering where R sn is proportional to I . This behavior can be qualitatively understood in the following manner. Consider the low flux limit of a single channel detector which obtains a series of N intensity measurements I i with a time delay δ t between measurements. For sufficiently low flux I i is either
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g 2 (δ t ) =
1 N ∑ Ii × Ii +1 N − 1 i =1 ⎛1 N ⎞ ⎜⎝ N ∑ I i ⎟⎠ i =1
2
(30)
The only contributions to the sum in the formula for g 2 appear when both I i and I i + 1 are non-zero. If M is the number 2 of such pairs of non-zero detection events, then M ~ I and the uncertainty, since the occurrence of such pairs is random, obeys counting statistics so that δ M = M ∝ I . There are a number of parameters which can be varied to optimize R sn . These include the energy bandpass of the monochromator, the size of the defining slits in front of the sample, and the angular size of the detector slits or the pixel size of an area detector. The optimal conditions will depend on details of the experiment such as whether a point detector or area detector is used, the susceptibility of the sample to damage, and the size and energy response of the detector. Optimizations for various conditions are discussed in the literature.[16b] We consider here two limits which commonly occur. For the case of a single element point detector, R sn is optimized for the pixel’s solid angle and the sample slits as far open as possible. This yields a very low value of contrast however, and at some point systematic errors limit the minimum acceptable value of the contrast. In practice a typical compromise of β: 1%–10% is often used. The case for a multi-element area detector is considered in detail in Falus et al.[16b] For this situation, R sn is maximized when the solid angle subtended by a detector pixel approximately matches the solid angle of a speckle. It is also interesting to solve Equation (29) for the shortest value of τ min that can be observed with a fixed scattering inten−2 sity I . In this case it can be seen that τ min ∝ I . Thus a doubling of the incident X-ray intensity allows access to four times faster time scales. This scaling implies that XPCS measurement will benefit strongly from the improvements planned in the next generation of synchrotron sources. Recently advances in X-ray optics have made it possible to manipulate the characteristics of speckle patterns by focusing the X-ray beam onto the sample. There are two main motivations for X-ray focusing. Firstly the transverse coherence length is typically much larger in the vertical direction than the horizontal direction for a third generation synchrotron source. For example at the Advanced Photon Source the vertical coherence length is of order 200 μm as compared to a horizontal coherence length of order 5 μm. If the vertical sample slit is opened up to this size (or even larger for optimal R sn ) the resulting speckle pattern can be too small to resolve. The size of the speckle pattern is on the order of the diffraction limit of the illuminated sample area which is given by Δ = Rd λ /W . Here Rd is the detector distance, and W the illuminated sample height in the direction of Δ. For example a 200 μm slit at 8 keV gives rise to a speckle size of 0.78 μm at 1m downstream of the sample. Even at 10 m, the speckle size would be too small to resolve with a typical CCD detector with pixel size of order 20 μm. While smaller pixel sizes are routinely available for
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optical CCD cameras, the quantum efficiency of such detectors for X-rays is typically very low since maintaining such high spatial resolution requires a thin depletion layer. An alternative method is to focus the X-ray beam so that the size of the focused beam on the sample is reduced and thus the size of each speckle is increased. Experiments have shown[32] that the coherent fraction of the beam (e.g. the ratio of the coherence length to the sample size) is approximately conserved. In this case the speckle sizes are magnified allowing for the speckle to be resolved and for the experiment to measure the full contrast of the speckle pattern. Examination of Equation (29) also suggests that there are advantages to focusing to sizes even smaller than is necessary to resolve the speckle. Since the intensity of detected scattering is proportional to the detector solid angle, and since the detector solid angle is limited to the speckle size (without reducing the measured contrast), decreasing the focus spot size increases I proportionally. The number of pixels measured, N , will also decrease proportionally, however, since R sn depends linearly on I but only as the square root of N the R sn will improve with the square root of the focus size. The gains that can be achieved by focusing are significant. For example, at sector 8-ID-I of the Advanced Photon Source the limited resolution of the area detector requires that the vertical slit size be limited to 20 μm in order to resolve the speckle pattern. Since the vertical coherence length at the sample is 200 μm this constraint forces 90% of the incident flux to be thrown away. Using a vertical focusing optics the incident beam can be focused to 20 μm so that the detector can resolve the speckles. This leads to an increase in R sn by 10. In fact, further focusing will increase R sn even though the total flux on the sample does not change. This is due to the increase in the size of the speckles at the detector. For example subsequent focusing from 20 μm to 2 μm will increase R sn by another factor of 10 . In practice these gains are partially offset by the finite efficiency of current optics, nevertheless significant improvements in R sn have been achieved.[32c] Another important avenue towards improved R sn in XPCS experiments is optimization of detectors. The initial introduction of area detectors for XPCS measurements provided a tremendous improvement in R sn over point detector measurements.[33] However, the initial cameras suffered from slow readout, low quantum efficiency and cumbersome data analysis. Subsequent development of camera hardware and data analysis software has improved the speed, efficiency and utility of these devices.[16a,34]
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one or zero photons. The first point in the correlation function g2 is then given by
3. Recent Studies using XPCS Fluid dynamics near surfaces have been of great interest not only for the fundamental understanding of the underlying physics and fluid mechanics but also for the practical design of complex fluids for engineering applications. A liquid surface can often be regarded as a smooth profile decorated with dynamic height variations of capillary waves, which arise from the collective hydrodynamic motion of the entire liquid elements.[35] Capillary waves on a liquid surface can be either propagating or over-damped depending on the wave length, surface tension and viscosity. The wave frequency is therefore
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a complex quantity ω + iΓ, ω and Γ being the propagating and damping frequencies, respectively. The propagating mode is unstable on highly viscous liquids such as simple liquids at low temperatures or polymer melts, leaving only the over-damped mode. Capillary waves play important roles in many surface and interface-related phenomena, such as extra roughening of the interface between immiscible polymer blends, spinodal dewetting and droplet coalescence.[36] Detection of the structure and dynamics of these capillary waves, on the other hand, provides a unique and powerful method for determining the rheological properties, and understanding other physical properties such as glass transitions in confined geometry and liquid-substrate interactions. For instance, the capillary wave relaxation time constant of viscous liquid films is proportional to the ratio of viscosity to the surface tension η/γ, thus providing an alternative measure of thin film viscosities.[6b,37] More recently, Sikorski et al. and Chushkin et al. measured the capillary wave relaxation rates of super cooled organic glass forming liquids of dibutyl phthalate and polypropylene glycol, respectively, and discovered strong evidence of surface induced elasticity.[38] In another study, Evans et al.[39] by measuring the top layer capillary wave relaxations, reported that the stiffening of the polystyrene film depends on the modulus of the underlying substrates near the glass transition temperature (Tg + 9 °C) even for very thick films, implying a strong effect of the substrate on the film over a substantial length scale even exceeding both the cooperative length of segmental mobility and the radius of gyration. In this review, we will briefly go through some historical but noteworthy experimental studies using surface XPCS and discuss recent progress on the dynamical study of surfaces of a variety of systems including bulk liquids, liquid crystals, thin films and polymer/nanoparticle composites. As discussed in Section 2(b), one of the important advantages of XPCS compared to DLS is that by performing XPCS at the grazing-incidence geometry one obtains the surface sensitivity to distinguish the dynamics from the surface and the bulk. This is achieved by choosing appropriate incident angles, i.e., below the critical angle for total external reflection, the evanescent wave only penetrates a few nanometers into the surface and therefore only fluctuations of the surface region are probed. If the incident angle is chosen such that X-ray standing waves are generated within the film of interest, scattering from a certain depth of the film, rather than the surface, dominates the total scattering,[40] therefore providing information about dynamics in the interior of the film. (a) Surface dynamics of low viscosity liquids On the surface of a low viscosity liquid or for very small wave vectors, the energy dissipation rate due to the friction between liquid elements is usually not high enough to prevent wave propagation, i.e, ω is finite. Both propagating and damping modes are present. Their frequencies are usually as high as kHz, which requires very fast detection (usually a point detector such as an avalanche photodiode running at several MHz) and sufficient coherent flux. Therefore, the XPCS experiment is often performed in the heterodyne mode by mixing a static reference beam in order to increase the signal-to-noise ratio, however, at a cost of coherence contrast.
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Figure 5. (a) Autocorrelation function g2 of a liquid water surface at 5 °C and wave vector q|| = 3.4 × 10−6 Å−1. Solid line is the fitting to the heterodyne correlation function. (b) Measured dispersion relation of the propagating capillary waves fitted to ω = (γ q||3 / ρ )1/2 . Inset shows corresponding damping constants. Reproduced with permission.[5b] Copyright 2003, American Physical Society.
In grazing-incidence geometry, specular reflection often serves as the reference beam.[14d] The heterodyne mixing in surface XPCS was first observed by Gutt et al.[5b] from a liquid water surface, where both the first and second order correlation functions are present in the intensity-intensity auto-correlation function I(q , t )I(q , t + τ ) t ~ 2I s Ir Re [ g 1 (q , τ )] + I s2 g 2 (q , τ )
(31)
with g 1 (q , τ ) = cos(ωτ )exp( −Γτ ) and g 2 (q , τ ) = exp( −2Γτ ) (Figure 5(a)). The measured dispersion relation is well fitted to the theoretical predictions for an ideal incompress1 /2 ible liquid ω = ( γ q ||3 / ρ ) [22] (Figure 5(b)). Although the damping constant is higher than the prediction Γ = 2ηq2 / ρ2 using the bulk viscosity, it does exhibit the required q|| dependence. The dynamic scattering function S(q,ω) for a simple incompressible liquid can be obtained from linear response theory.[23] A critical wave vector, q||,c = 4γρ/5η 2, can then be deduced
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have to be solved numerically and fit the experiment data very well throughout the entire q range (Figure 6(b)). Gutt et al.[44] have studied the XPCS from partially wetting liquid films on relatively smooth substrates such as silicon and glass. The liquids were n-hexane and cyclohexane introduced at vapor pressures well below saturation onto slightly cooled substrates. They found that very thin films (6 nm thickness) conformed to the substrate while thicker films (10 nm to 60 nm thickness) showed characteristic q−3 power-law dependence for the off-specular diffuse scattering, seen previously for partially wetting films. Interestingly, the coherent scattering showed no dynamics at all, as evidenced by static speckle patterns over time scales from 1s to 20 mins. They ascribed this puzzling result to the breakdown of the classical hydrodynamic theory for the capillary fluctuations on liquid films for partially wetting films, probably due to different boundary conditions.
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from S(q,ω)[41] so that beyond this wave vector the capillary waves are over-damped with Γ = 2η /γ q .[42] This is essentially the asymptotic limit of equation (18) as the film thickness approaches infinity. When q
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X-ray Photon Correlation Spectroscopy Studies of Surfaces and Thin Films Sunil K. Sinha,* Zhang Jiang, and Laurence B. Lurio
changing phase differences between the scattered waves. It was recognized about 50 years ago that this dynamical fluctuation of brightness on a given area of the detector could yield information about the dynamics of the object. Thus with the advent of highly coherent laser beams, this method, i.e., Dynamical Light Scattering (DLS), or Photon Correlation Spectroscopy (PCS), became well established[2] as a method for studying the diffusion of colloids or entities such as proteins in solution. One of the main applications of this technique was to obtain a quick method of obtaining particle sizes, since the diffusion coefficient for a dilute solution of colloids, etc. is related to the hydrodynamic radius of the particle, the viscosity of the solution and the temperature via the StokesEinstein relation.[3] DLS yields dynamical information at fairly small values of the wave vector transfer q (i.e. rather large length scales), being limited to dimensions greater than the wavelength of visible light. While it was recognized that carrying out similar experiments with X-rays would make it possible to study larger values of q (i.e. shorter length-scales), such experiments were not possible due to the relatively low brilliance of available X-ray sources, until the advent of high-brilliance synchrotron X-ray sources, beginning with the so-called third generation light sources. The high brilliance of these and subsequent sources have made it possible to obtain a sufficiently high flux of coherent photons from such sources (in a manner to be discussed below) to be able to carry out photon correlation spectroscopy studies using coherent (or partially coherent) beams of X-ray photons, thus giving rise to the technique of X-ray Photon Correlation Spectroscopy (XPCS). The technique of XPCS is accessible at a number of coherence beamlines at synchrotron facilities worldwide, among which are a few dedicated ones such as Sector 8-ID-I at the APS (Advanced Photon Source, USA), ID10A Troika I at the ESRF (European Synchrotron Radiation Facility), P10 at PETRA III, DESY (Germany), as well as the newly constructed CHX at the NSLS II (National Synchrotron Light Source II, USA). Within little more than a decade, experiments using this technique have gone from being simple demonstrations of a novel technique to actually being used to obtain information difficult to obtain by other means. XPCS has had a particularly significant impact on soft condensed matter research, such as the study of the dynamical
The technique of X-ray Photon Correlation Spectroscopy (XPCS) is reviewed as a method for studying the relatively slow dynamics of materials on time scales ranging from microseconds to thousands of seconds and length scales ranging from microns down to nanometers. We focus on the application of this technique to study dynamical fluctuations of surfaces, interfaces and thin films. We first discuss instrumental issues such as the effects of partial coherence (or alternatively finite instrumental resolution) and optimization of signal-to-noise ratios in the experiments. We then review what has been learned from recent XPCS studies of capillary wave fluctuations on liquid surfaces and polymer films, of nanoparticles used as probes to study the interior dynamics of polymer films, of liquid crystals and multilamellar surfactant films, and of metal surfaces, and magnetic domain wall fluctuations in antiferromagnets. We then discuss studies of non-equilibrium dynamics described by 2-time correlation functions. Finally, we briefly speculate on possible future XPCS experiments at new synchrotron sources currently under development including studies of dynamics on time scales down to femtoseconds.
1. Introduction When a coherent beam of light falls on an object with any type of disorder (static or dynamic), the scattered light is broken up into bright and dark regions as seen in a 2D detector or on a screen. These are due to the irregular phase differences and hence irregular interference between the waves scattered from different parts of the object, and this apparently random array of bright spots is termed a “speckle pattern”, a phenomenon that has been known for years.[1] If different parts of the object fluctuate with time, the speckle pattern also fluctuates due to the Prof. S. K. Sinha Dept. of Physics University of California San Diego 9500 Gilman Drive, La Jolla, CA 92093–0319, USA E-mail: [email protected] Dr. Z. Jiang Advanced Photon Source Argonne National Laboratory Argonne, IL 60439, USA Prof. L. B. Lurio Dept. of Physics Northern Illinois University DeKalb, IL 60115, USA
DOI: 10.1002/adma.201401094
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Dr. Sunil K. Sinha is Distinguished Professor of Physics in the Department of Physics at the University of California, San Diego. He received his Ph.D. in Physics from Cambridge University. He has served as Group Leader in Neutron Scattering at Argonne National Laboratory, Group Leader of X-ray Scattering at Brookhaven National Laboratory, Senior Research Associate at Exxon Corporate Research Laboratories, and Associate Division Director at Argonne’s Advanced Photon Source. His group’s research is concerned with studying the structure and dynamics of condensed matter using the techniques of x-ray and neutron scattering.
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behavior of colloids,[4] liquids,[5] polymers[6] and clays.[7] This is not surprising since, in some sense, what makes soft matter “soft” is its ability to exhibit significant thermal fluctuations near room temperature. However, the impact of XPCS has not been limited to this area, and has also been used in areas such as the study of magnetic domain fluctuations,[8] phase ordering in glasses[9] and metal alloys,[10] atomic diffusive motion in crystals,[11] martensitic phase transformations[12] and the ordering of single crystal interfaces.[13] There already exist several excellent reviews dealing with XPCS in the literature.[14] In this article we will briefly describe the technique and related instrumentation, and discuss recent applications of this technique to surfaces, interfaces and thin films. The recent development of new sources with a much higher degree of coherence, such as the X-ray free electron laser sources (XFEL’s) or the planned diffraction-limited storage ring (DLSR) sources will make it possible to greatly extend both the timescales and length scales which can be studied with XPCS, and we will briefly discuss prospects for future experiments with such sources later in this article. Techniques such as DLS, XPCS and neutron spin echo probe the sample scattering function in real time, and as such are complementary to inelastic scattering (of light, X-rays or neutrons) which probe the scattering function in frequency space. For slow dynamics the former are more convenient as the energy resolution required for inelastic scattering becomes too stringent. Interestingly, the results of DLS experiments are often Fourier transformed (by using the power spectrum of the scattered light, e.g. using a spectrum analyzer) to give the scattering function as a function of frequency.[15] The timescales which can be studied with XPCS are primarily limited by the flux of coherent photons falling on the sample, since the ability to resolve short-time dynamics relies on sufficient flux being scattered into a speckle within the time-scale of interest. A figure of merit that has been used in dynamic light scattering is that an experiment requires at least one scattered photon per speckle over the timescale of interest. In fact, this metric is too conservative, since it has been shown that with the use of area detectors many speckles can be measured at once, with a concomitant increase in signal.[16] However, if an area detector is used, it must be able to read out within the relevant timescale and the available readout speed of existing x-ray area detectors then becomes the limiting factor in many cases. Current XPCS experiments have studied dynamics on time scales from microseconds to thousands of seconds. In this sense they are complementary to neutron spin echo experiments which probe dynamics of systems in real time on time scales of nanoseconds and shorter. The wavevector transfers (q values) which can currently be studied by XPCS can range from 10−3 nm−1 to several nm−1. (See Figure 1). The transverse coherence length of a photon beam can be interpreted in terms of the familiar Young’s double slit experiment (see Figure 2(a)) in terms of how far apart the two slits can be before the visibility of the fringes due to the photon beam on the far screen drops below some given criterion, i.e the characteristic distance in the direction transverse to the beam over which there is phase coherence of the beam. There are evidently two such transverse coherence lengths and they are given by the formulae
Dr. Zhang Jiang is currently a scientist at the Advanced Photon Source, Argonne National Laboratory. He received a B.S. in applied physics from University of Science and Technology of China in 2001 and a Ph.D. in physics from University of California, San Diego in 2007. Since then he has been working at the Advanced Photon Source, where he is responsible for the operation, development and maintenance of the X-ray photon correlation spectroscopy and grazing-incidence X-ray scattering beamlines. His scientific research mainly focuses on using coherent X-ray scattering and grazing-incidence surface scattering to study the structure, kinetics and dynamics of liquids, soft matter, and nanocomposite materials. Dr. Laurence Lurio is a professor at Northern Illinois University and chair of the physics department there. He received a B.A. in physics from Columbia University and a Ph.D. in physics from Harvard University. He was part of the team which constructed and commissioned the dedicated XPCS beamline at sector 8-ID of the APS, and is currently a member of the beamline access team planning the first experiments at the coherent hard x-ray (CHX) beamline at NSLS-II. He has also served as chair of the APS User Organization Steering Committee. His research interests focus on structural and dynamical properties of condensed matter with a particular emphasis on liquids, polymers and biomaterials.
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pattern, while the speckle intensity will diminish as the pinhole size is increased beyond that size. There is also a limitation on the longitudinal coherence, which is governed by the monochromaticity of the beam. The longitudinal coherence can be related to the maximum difference in the length of the arms of a Michelson interferometer which can be allowed and still obtain an interference pattern. It is given by ⎛ λ ⎞ ξl = λ ⎜ ⎝ Δλ ⎟⎠
Figure 1. Illustration of regions of frequency-wave vector space covered by various dynamical probes, including Inelastic Neutron Scattering (INS), Inelastic X-Ray Scattering (IXS), Neutron Spin Echo, Nuclear Forward Scattering (NFS), Photon Correlation Spectroscopy, i.e., Dynamical Light Scattering (PCS) and X-ray Photon Correlation Spectroscopy (XPCS). Reproduced with permission.[14d] Copyright 2008, Springer.
ξx =
λR λR and ξ y = 2πσ x 2πσ y
(1)
where λ is the average photon wavelength and R is the distance to the source. Here the source is assumed to have an incoherent Gaussian intensity profile of the form I( x , y ) ∝ exp ( − x 2/2σ x2 − y 2 /2σ y2 ). A beam passing through a pinhole of dimensions ξxξy will yield a well-defined speckle
Figure 2. (a) Interference pattern arising from a Young’s double slit experiment with a finite source size. Rays originating from the top and bottom of the source produce offset interference patterns which limits the maximum transverse separation of the slits which can still yield an interference pattern. (Colors are for clarity and not meant to imply differing frequencies). (b) The longitudinal coherence length ξl can be related to the width of the wave packet made up of a small distribution of wavelengths over which interference is possible.
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(2)
where Δλ is the wavelength spread of the beam (see Figure 2(b)). This is typically defined as the full width at half maximum of the wavelength spread of the beam, but the precise longitudinal coherence length will depend on the functional form of the wavelength spread. The transverse coherence lengths of the radiation actually incident on the sample will also be affected by X-ray optics, slits, etc. in the beam path prior to the sample. The product of these three lengths is proportional to the coherence volume of the beam, i.e. the volume within which the scattering is coherent. The brightness B of a source is defined as the number of photons emitted per second per unit source area per unit solid angle. Thus the number of transversely coherent photons passing per unit time through a pinhole of dimensions ξxξy is proportional to I coh =
π 2Bσ x σ y ξx ξy λ 2B = R2 4
(3)
However, if the source is not highly monochromatic (e.g. a particular harmonic of an undulator at a 3rd generation source), then ξl will be quite small. To increase it, one needs to monochromate the beam further, for example using a crystal monochromator (where Δλ/λ can be ∼10−4), so that the number of coherent photons will then be proportional to the spectral brightness (sometimes referred to as brilliance) of the source, which is defined as the brightness per 0.1% bandwidth in energy. For current third generation undulator sources, the transverse coherence lengths are of the order of 5 μm to 200 μm in the vertical direction and 2 μm to 5 μm in the horizontal direction while the longitudinal coherence length can be 1–2 μm after monochromatization. The effects of partial coherence will be discussed in more detail below. In general, the scattering from 2 points separated by a distance Δr can interfere coherently if Δr is less than both (not “either” as stated in Ref. [17]) ξt /sin α i and k0 ξ1 /q where αi is the angle the incident beam makes with Δr,k0 is the magnitude of the incident wavevector and q is the component of the wavevector transfer parallel to Δr[17] (see Figure 3). Thus we see that the effective transverse coherence length in the plane of incidence is greatly magnified if αi is small (e.g. if we have scattering from a surface under conditions of grazing incidence) and the effective longitudinal coherence length is magnified for small q values, or for scattering from a surface in directions close to specular. If the coherence volume is much greater than the sample volume, then all points in the sample scatter coherently, and
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Figure 3. Illustration of the maximum separation of two points on a surface in order to produce coherent scattering. Transverse coherence requires Δr < ξt /sinα i , while longitudinal coherence puts a maximal limit on the X-ray path difference (bold line segments), i.e., Δr < k0ξl /q|| with q|| = k0 ( cos α i − cos α f ) the in-plane component of the wavevector transfer.
the speckle pattern is determined by the exact positions of all scatterers in the sample, i.e. it represents a “fingerprint” of the sample. If the beam is only partially coherent, the visibility of the speckle pattern decreases. A recent discussion of the statistical properties of the intensities of speckles and their visibility has been given by Grubel, Madsen and Robert,[14d] who show that the speckle contrast goes to zero as the coherence volume decreases relative to the scattering volume. The dynamics of a system is usually characterized in terms of the dynamical structure factor or scattering function S(q,ω) which is given by S( q , ω ) =
1 2π
∞
∫ dte
− iω t
ρ( q ,0)ρ( − q , t )
(4)
−∞
where ω is the energy loss of the probe, and ρ(q,t) is the Fourier component of the electron density at time t. The average is either a statistical or quantum mechanical average over the system. This is the function which is measured in conventional inelastic scattering experiments. The frequency Fourier transform of S(q,ω) is the function g1(q,t), also known as the intermediate scattering function. This is given by g 1 ( q , t) =
1 ρ( q ,0)ρ( − q , t ) N
(6)
where C is a constant related to parameters such as the incident beam intensity, etc. and the average is evaluated at time t. Intensity correlation experiments are of two categories: (a) homodyne experiments where the time variation of the scattered beam alone is measured and (b) heterodyne experiments, where the scattered beam is made to interfere with a static reference signal. Most XPCS experiments are carried out in the homodyne mode, although this is by no means true of DLS.[18] What is measured in a homodyne XPCS experiment is the intensity autocorrelation function given by
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(7)
where the average is over the time t′. Note that as written g2(q,t) is fourth order in the electric fields, since the intensities are each second order in the fields. One can often write the scattered electric fields (which are proportional to ρ(q,t) as Gaussian random variables with zero mean. Then the above equation can be decomposed into a simpler product of correlation functions of electric fields, rather than intensities i.e. g 2 ( q , t) = 1 + F( q , t)
2
(8)
where F( q , t ) = g 1 ( q , t ) / g 1 ( q ,0) is the normalized intermediate scattering function. This is known as the Siegert Relation.[2] Note that in this case the phase of g1(q,t) is undetermined. If the beam is only partially coherent, so that the speckle contrast is low, and the length scales associated with the sample structure are significantly smaller than the illuminated sample dimensions, then this relation is often modified[19] to read g 2 ( q , t) = 1 + β ( q ) F( q , t)
2
(9)
where the q-dependent contrast factor β(q) depends on the incident beam properties and the experimental setup. It can in principle vary from 0 to 1. We postpone discussion of this quantity and the discussion of the effects of partial coherence and resolution effects to Section 2. The dynamics of the system in terms of the intermediate scattering function can be inferred from g2(q,t) –1. A heterodyne experiment in principle can be used to deter2 mine both the real part of g1(q,t) and g 1 ( q , t ) . For details see Refs. [18,20] The optimization of experiments at 3rd generation sources is often governed by the need to balance the requirement for a high counting rate to obtain better statistics for the function g 2 ( q , t ) in a given time, and the requirement for better contrast, which often requires lower absolute beam intensity.
(5)
where N is the total number of scatterers. For a completely coherent X-ray beam, the scattering intensity from the sample for a given q (no energy analysis) at any instant of time t is given by I ( q , t ) = C ρ( q , t ) ρ( − q , t )
2
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g 2 ( q , t ) = I( q , t ′)I( q , t ′ + t ) / I( q )
2. Methods We now discuss briefly some of the general types of studies carried out with XPCS. (a) Transmission XPCS experiments and colloidal diffusion While such experiments are not the main subject of the present review, we mention them in passing for historical reasons, as the earliest XPCS experiments were carried out on colloidal particles in solution.[4] These experiments are carried out in transmission geometry as in small angle X-ray scattering experiments. What is probed is the dynamics in the bulk of the sample, such as the diffusion of particles in solution. For a system of discrete particles the Fourier component of the electron density is given by ρ( q , t ) = ∑ f i (q)e − iq . r ( t ). Here f i is the form factor associated with particlei i . In the simplest case where all particles have the same size one obtains:
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g 1 ( q , t) =
f 2(q) N
2
∑e
− iq . ri (0) iq . rj ( t )
e
(10)
i, j
In a system where the particles are undergoing random Brownian motion and where their motion is uncorrelated, the averages for i ≠ j and finite q vanish and we are left with g 1 ( q , t) =
f 2(q) ∑ e − iq .[ri (0)− ri ( t )] N i
(11)
Using the fact that the ri are Gaussian random variables, we can 2 1 − {q .[ ri (0) − ri ( t )]} f 2(q) e 2 write g 1 ( q , t ) = and using the diffusion ∑ N i 2 law (ri (0) − rj (t )) = 6Dt and averaging over all orientations of q and [ ri (0) − ri (t )] , we obtain the well-known result for hydrodynamic diffusion 2
F( q , t ) = e − Dq t
(12)
Thus for hydrodynamic diffusion, the relaxation time is 1/Dq2. As mentioned previously, this is often used in DLS to determine the diffusion constant of particles in solution, and hence, using the Stokes-Einstein relation, to obtain an estimate of the hydrodynamic radius of the particle. For interacting particles the form of the dynamic structure factor is generally more complex. However, one can attempt to maintain connection to this formalism by replacing D with a q-dependent function D(q,t).[19] (b) XPCS at Grazing Incidence and dynamics of liquid surfaces The most straightforward application of XPCS to the study of liquid surfaces is to carry out measurements using an X-ray beam which impinges on the surface at grazing incidence. For a free liquid-vapor interface what one typically measures in such cases is the dynamics of capillary wave fluctuations. The first such measurements were carried out by Seydel et al.[5a] on liquid glycerol and by Kim et al.[6b] on molten polymer films. There have also been many earlier DLS studies of capillary wave fluctuations on liquid surfaces.[15] For a liquid surface, the expression for the diffuse scattering cross-section at an instant of time t [21] yields I( q , t ) =
PA − q2zσ 2 e q z2
∫ ∫ dXdY {e
q 2z C ( R , t )
− 1}e − i [ qx X + qyY ]
(13)
Here P is a constant related to the incident beam intensity and the electron density of the liquid, A is the illuminated surface area, σ2 is the mean square roughness, R is the in-plane vector (x, y), and C(R, t) is the time dependent height-height correlation function C( R , t ) = u z ( r , t )u z ( r + R , t )
r
(14)
where u z ( r , t ) is the displacement of a point on the liquid surface at in-plane position r and time t. For small values of qz, the exponential in the integral can be expanded, leading to the approximate expression
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I( q , t ) = PAe − qzσ
2
∫ ∫ dX dYC(R , t)e
− i [ qx X + qy Y ]
(15)
The measured g2 function, using the Siegert relation, yields the corresponding g1(q,t) function: 2
2
g 1 ( q , t ) = P ′ e − qzσ u z ( q ,0)u z ( q , t )
(16)
where the correlation function is between in-plane Fourier components of uz(r, t). The vector q indicates the in-plane component of q. For liquids with low viscosity, such as water, we −Γ t have propagating capillary waves, so Re[ g 1 (q, t )] ~ cos (ω q t ) e q . The frequencies and damping constant are given approximately by[22]
ω = ( γ q ||3 / ρ )
1/2
and Γ = 2η q 2 /ρ .
(17)
Here γ is the surface tension, ρ the mass density, and η is the viscosity. For highly viscous liquids, such as polymer melts, the capillary waves are overdamped, so there is no cos (ω q t ) factor in Re [ g 1 (q, t )] . In this case homodyne XPCS experiments are sufficient to determine the damping constant and its q-dependence. Such experiments have been carried out for several molten polymer films, and will be discussed in more detail in the next section. The hydrodynamic theory of capillary fluctuations of a thin viscous liquid film on a solid substrate has been worked out by Jackle,[23] and by Harden, Pleiner and Pincus.[24] From their general formulation, Kim et al.[6b] derived the following expression for the relaxation time τ q (or Γ q−1 ) for overdamped capillary waves in a thin film of thickness h, surface tension γ and viscosity η:
τq =
2η[cosh 2 (qh ) + q2h 2 ] γ q [sinh(qh )cosh(qh ) − qh ]
(18)
From Equation (18), it may be seen that τ q / h is a function of only the dimensionless product qh and is proportional to the ratio η / γ . This scaling relation has been verified using grazing incidence XPCS for a number of polystyrene films of various thicknesses at several temperatures.[6b,25] If the liquid is viscoelastic rather than purely viscous, we shall see in Section 3(b) how Equation (18) above has to be modified to fit the data. Another advantage of grazing incidence scattering from thin films or surfaces is that by varying the grazing angle of incidence from below to above the critical angle for total reflection, and thus changing the penetration depth of the X-rays into the film, one can selectively probe the surface or underlying regions of the film. This method was employed by Hu et al. to study dynamics of buried interfaces in a polymer bilayer system.[26] (c) Effects of Partial Coherence and Resolution The measured intensity I( q ) in a detector in an actual experiment involves not just the intermediate scattering function g 1 ( q , t ) but also the partial coherence, characterized by the Mutual Coherence Function (MCF)[27] of the beam incident on the sample, as well as the finite detector acceptance angle. For beams which are not perfectly coherent and monochromatic,
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U α ( r , t ) = Aα ( r , t )exp( − iω t )
(19)
for the intensity in the detector (where we omit the time variable): 2
⎛ e2 ⎞ 1 I(q) = I 0 ⎜ 2 ⎟ 2 ∫ d Δq ⊥ P( Δq ⊥ ) ⎝ mc ⎠ λ
∫ ∫ drdr ′ ρ( r )ρ( r ′)e
iq ⋅(r − r ′ )
Γ( r⊥ , r ′ ⊥ , q ⋅ ( r − r ′)/ω )
where Δq ⊥ is the component of q perpendicular to the average scattered beam direction due to the finite size of the overall detector element and P( Δq ⊥ ) is the acceptance function of the detector (usually represented by a Gaussian function). Substituting for the MCF using Equations (21) and (22), we obtain:
where α is the polarization index, r denotes a point in space, and ω is the average frequency, while the amplitude represents a slowly varying (relative to the timescale of ω ) time-dependent function. The MCF is then defined by
⎛ e2 ⎞ 1 I(q) = I 0 ⎜ 2 ⎟ 2 ∫ d Δq ⊥ P (Δ q ⊥ ) ⎝ mc ⎠ λ
Γ α ( s, s′, τ ) = Aα ( s, t )Aα* ( s′, t + τ )
where
(20)
where s, s′ denote points in a plane perpendicular to the average beam propagation direction and the average represents a time average over many phase fluctuations of the radiation (which for X-rays have periods of 10−15 to 10−16 seconds).We shall henceforth assume completely plane polarized radiation and drop the polarization index α. In principle the MCF will contain the Fresnel diffraction effects from the slit prior to the sample, but for simplicity, we here assume a Gaussian-Schell Model[29] form for the MCF : Γ( s , s ′, τ ) = I 0 Ψ( s)Ψ * ( s′) g ( s − s′)F(τ )
(21)
where Ψ( s) represents an amplitude at s (which can be cut off if the beam cross-section is smaller than the sample dimensions) which is normalized to unity for maximum amplitude, I 0 is the incident beam intensity, and g(s − s′) and F(τ) can be represented by g (x , y) = e − x F(τ ) = e −τ /τ l
2
2 2 /2ξ x2 − y /2ξ y
e
(22)
where ξx and ξy are the transverse coherence lengths in the x, y directions respectively, and the coherence time τl is determined by the monochromaticity of the beam. Let us choose a plane in the sample perpendicular to the incident beam and choose the origin in the plane, so that the MCF at the sample position can be defined by Equations (21) and (22), and chose the direction of propagation of the incident beam as the z axis. We use the kinematic (HuygensFresnel) theory of wave propagation to describe the scattering as in Ref.[17] but here we derive a simplified version that describes the scattering in terms of the MCF at the sample instead of at the incident aperture, and neglect the higher order Fresnel terms (which is a good approximation if the correlation lengths of the scattering elements are small and the detector area is small –which is true if we are considering single pixels or small groups of pixels). This yields
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(23)
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this will result in both a loss of contrast (expressed in terms of the contrast factor β ( q ) in Equation (9) and smearing in q-space of the measured g 1 ( q , t ) function. Quantitative studies of the X-ray contrast and comparisons of experiments with theory have been made by several authors.[28] Those readers who are not interested in the details discussed here may skip to Section 2(d) below. The MCF is determined by the source and all the optical elements (including slits) prior to the sample and can be characterized in terms of the 2 transverse coherence lengths and the longitudinal coherence length. The complex analytical signal describing the incident electric field is given by
2
∫ ∫ d r d r ′ρ ( r ) ρ ( r ′ ) e
iq ⋅( r − r ′ )
R( r , r ′ )
(24)
R( r , r ′) = Ψ( r⊥ )Ψ * ( r⊥′ ) 2
2
e −( x − x ′ ) /2ξx e −( y − y ′ ) /2ξy e 2
2
− q ⋅( r − r ′ ) /k0 ξl
.
(25)
Here we have written ωτ 1 as k0 ξl to express it in terms of the longitudinal correlation length. R( r , r ′ ) acts like a resolution function which limits the coherent scattering from points separated by more than the transverse coherence length in the direction perpendicular to the incident beam and by more than k0 ξl in the direction along q . As seen from Equation (25), the longitudinal coherence length will not play a limiting role at small q for grazing incidence surface scattering, or if q is close to the specular direction. For small angle scattering q ⋅ ( r − r ′ ) is very small for r − r ′ along the incident beam, which implies the sample can be extended in this direction without loss of contrast. The longitudinal coherence length ξl has the property of limiting correlations contributing to scattering along the direction of q, resulting in streaking of the speckle patterns along q as we go to larger q. As an example, Figure 4(a) shows a speckle pattern from an aerogel sample measured using an X-ray beam monochromatized with a Ge-111 crystal with a bandwidth of ΔE /E = 3.3 × 10 −4 . The speckles appear to be completely isotropic. By contrast, the aerogel scattering pattern measured in Figure 4(b) was obtained using a pink beam (the direct undulator harmonic, without any monochromatization). The bandwidth for a pink beam is of order ΔE/E = 0.026. In this case the speckles are highly extended along the radial direction. A typical longitudinal coherence length ξl is, for example, ∼2 μm for a monochromatized beam of 8 keV with a bandwidth of ∼10−4. This limits experiments to very thin samples at high scattering angles. However, for small angle scattering, we can use samples as thick as several mm, before the effects of longitudinal coherence become significant. This has been noted by Sandy et al.[28c] Equation (24) yields the scattering from the exact electron density configuration at a given time, resulting in a speckle pattern, while the function R ( r , r ′ ) smears out this pattern in
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g 2 ( q ,τ ) = 1 +
2
⊥
P( Δq ⊥ ) ∫ d q ′ g 1 ( q ′, τ ) R 2 ( q − q ′)
∫ d q ′ g ( q ′,0)R( q − q ′)
2
(28)
1
Figure 4. Speckle patterns from (a) monochromatic incident X-ray beam, and (b) X-ray beam with broad energy bandwidth. Reproduced with permission.[28c] Copyright 1999, Intl. Union of Crystallography.
q-space, resulting in decreasing speckle visibility for smaller coherence lengths. It has been shown (see Refs. [14d,28a] that the speckle visibility is reduced by the number of coherence volumes contained in the illuminated sample volume. The integration over the finite solid angle of the detector (or group of pixels in the 2D detector) further smears the observed speckle pattern. For an actual measurement of I( q ) we average ρ( r )ρ( r ′) over time. For many systems which are ergodic and translationally homogeneous (such as fluids), this average is a function of (r-r′) alone, and if we assume constant values for the Ψ( r⊥ ) Ψ*(r′⊥) over the sample, then R ( r , r ′ ) can also be written asR ( r − r ′ ) . If its Fourier transform is R(q), then we have 2
⎛ e2 ⎞ 1 I( q ) = I 0 ⎜ 2 ⎟ 2 ∫ d Δq ⊥ P( Δq ⊥ ) ⎝ mc ⎠ λ
∫ dq
′
(26)
ρ( q ′ ) ρ * ( q ′ ) R( q − q ′ )
I( q , t )I( q , t + τ ) = I( q )
2
4
⎛ e2 ⎞ 1 + I 02 ⎜ 2 ⎟ 4 ⎝ mc ⎠ λ
∫ d Δq
2
⊥
P( Δq ⊥ ) ∫ d q ′ g 1 ( q ′, τ ) R 2 ( q − q ′)
(27)
(We assume here that the time correlations and the intensity normalization are performed for individual pixels and then averaged over the detector area). Using Equations (7), (26) and (27), we thus get the result for the effective g2 function including partial coherence and instrumental resolution effects as
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This expression is more complicated than the simplified expression (Equation (9)) commonly used, but shows the same behavior, namely a decrease of the second term on the right from the “ideal” value of |F(q,τ)|2 roughly in the ratio of the coherence volume to the total scattering volume. A discussion in terms of treating the speckles counted in the detector as “independent modes” by Lemieux and Durian[30] with speckle intensities I j (j = 1 to N) where N speckles are counted in the detector leads to a simpler result which has some similarities to Equation (28), namely g2 is given by Equation (9) with the parameter β given by N N 2 ⎛ ⎞ β = ∑ Ij /⎜∑ Ij ⎟ ⎝ j =1 ⎠ j =1
2
An interesting case arises when the function g1(q, τ) contains a static component (e.g. specular reflection in the case of surface XPCS, or a Bragg reflection) at a q-value close to what is being nominally measured, which contributes to the integrals in Equation (28). This then provides a heterodyne component to the measurement of g 2 ( q , τ ). This may change the form of g 2 ( q , τ ) from the true g 2 ( q , τ ). This has been discussed in detail in Ref. [87] for the case of propagating capillary waves, where by considering the effects of partial coherence, the specular reflection and detector acceptance (including the Fresnel factors neglected in the above derivation) it was shown that one can achieve a cross-over from cos2(ωτ) behavior to cos(ωτ) behavior in the measured correlation function, and the damping constants can be significantly different. For highly overdamped waves, the effect however is fairly small.[5b] (d) Focusing and Signal to Noise Ratios in XPCS
Equation (26) explicitly shows the smearing in q-space as a convolution of the true scattering function with an effective “resolution function”. For XPCS measurements, we are interested in the average I( q , t )I( q , t + τ ) over time t , as shown in Equation (7). Assuming the scattered electric field at the detector is a Gaussian random variable with zero mean, we can apply the usual decoupling scheme used to derive the Siegert relation, Equation (9). We omit the details of the calculation here, but the result is
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It is crucial in an XPCS experiment to optimize the signal to noise ratio due to the limited coherent flux available at current third generation synchrotrons as well as the need to reduce radiation damage to samples. A relation for the signal to noise ratio R sn in PCS in the limit of low intensity was given by Jakeman[31] and adapted to XPCS by Falus et al.[16b] This is given by: R sn = I β
TΔtN g2
(29)
Here I is the detector count rate (or count rate per pixel in the case of an area detector), β the contrast, Δt the integration time of the detected signal, T the duration of the experiment and N the number of independent detector channels averaged together to obtain the signal. A particularly interesting feature of R sn is that it depends linearly on the total incident intensity as compared with the usual expectation for incoherent X-ray scattering where R sn is proportional to I . This behavior can be qualitatively understood in the following manner. Consider the low flux limit of a single channel detector which obtains a series of N intensity measurements I i with a time delay δ t between measurements. For sufficiently low flux I i is either
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g 2 (δ t ) =
1 N ∑ Ii × Ii +1 N − 1 i =1 ⎛1 N ⎞ ⎜⎝ N ∑ I i ⎟⎠ i =1
2
(30)
The only contributions to the sum in the formula for g 2 appear when both I i and I i + 1 are non-zero. If M is the number 2 of such pairs of non-zero detection events, then M ~ I and the uncertainty, since the occurrence of such pairs is random, obeys counting statistics so that δ M = M ∝ I . There are a number of parameters which can be varied to optimize R sn . These include the energy bandpass of the monochromator, the size of the defining slits in front of the sample, and the angular size of the detector slits or the pixel size of an area detector. The optimal conditions will depend on details of the experiment such as whether a point detector or area detector is used, the susceptibility of the sample to damage, and the size and energy response of the detector. Optimizations for various conditions are discussed in the literature.[16b] We consider here two limits which commonly occur. For the case of a single element point detector, R sn is optimized for the pixel’s solid angle and the sample slits as far open as possible. This yields a very low value of contrast however, and at some point systematic errors limit the minimum acceptable value of the contrast. In practice a typical compromise of β: 1%–10% is often used. The case for a multi-element area detector is considered in detail in Falus et al.[16b] For this situation, R sn is maximized when the solid angle subtended by a detector pixel approximately matches the solid angle of a speckle. It is also interesting to solve Equation (29) for the shortest value of τ min that can be observed with a fixed scattering inten−2 sity I . In this case it can be seen that τ min ∝ I . Thus a doubling of the incident X-ray intensity allows access to four times faster time scales. This scaling implies that XPCS measurement will benefit strongly from the improvements planned in the next generation of synchrotron sources. Recently advances in X-ray optics have made it possible to manipulate the characteristics of speckle patterns by focusing the X-ray beam onto the sample. There are two main motivations for X-ray focusing. Firstly the transverse coherence length is typically much larger in the vertical direction than the horizontal direction for a third generation synchrotron source. For example at the Advanced Photon Source the vertical coherence length is of order 200 μm as compared to a horizontal coherence length of order 5 μm. If the vertical sample slit is opened up to this size (or even larger for optimal R sn ) the resulting speckle pattern can be too small to resolve. The size of the speckle pattern is on the order of the diffraction limit of the illuminated sample area which is given by Δ = Rd λ /W . Here Rd is the detector distance, and W the illuminated sample height in the direction of Δ. For example a 200 μm slit at 8 keV gives rise to a speckle size of 0.78 μm at 1m downstream of the sample. Even at 10 m, the speckle size would be too small to resolve with a typical CCD detector with pixel size of order 20 μm. While smaller pixel sizes are routinely available for
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optical CCD cameras, the quantum efficiency of such detectors for X-rays is typically very low since maintaining such high spatial resolution requires a thin depletion layer. An alternative method is to focus the X-ray beam so that the size of the focused beam on the sample is reduced and thus the size of each speckle is increased. Experiments have shown[32] that the coherent fraction of the beam (e.g. the ratio of the coherence length to the sample size) is approximately conserved. In this case the speckle sizes are magnified allowing for the speckle to be resolved and for the experiment to measure the full contrast of the speckle pattern. Examination of Equation (29) also suggests that there are advantages to focusing to sizes even smaller than is necessary to resolve the speckle. Since the intensity of detected scattering is proportional to the detector solid angle, and since the detector solid angle is limited to the speckle size (without reducing the measured contrast), decreasing the focus spot size increases I proportionally. The number of pixels measured, N , will also decrease proportionally, however, since R sn depends linearly on I but only as the square root of N the R sn will improve with the square root of the focus size. The gains that can be achieved by focusing are significant. For example, at sector 8-ID-I of the Advanced Photon Source the limited resolution of the area detector requires that the vertical slit size be limited to 20 μm in order to resolve the speckle pattern. Since the vertical coherence length at the sample is 200 μm this constraint forces 90% of the incident flux to be thrown away. Using a vertical focusing optics the incident beam can be focused to 20 μm so that the detector can resolve the speckles. This leads to an increase in R sn by 10. In fact, further focusing will increase R sn even though the total flux on the sample does not change. This is due to the increase in the size of the speckles at the detector. For example subsequent focusing from 20 μm to 2 μm will increase R sn by another factor of 10 . In practice these gains are partially offset by the finite efficiency of current optics, nevertheless significant improvements in R sn have been achieved.[32c] Another important avenue towards improved R sn in XPCS experiments is optimization of detectors. The initial introduction of area detectors for XPCS measurements provided a tremendous improvement in R sn over point detector measurements.[33] However, the initial cameras suffered from slow readout, low quantum efficiency and cumbersome data analysis. Subsequent development of camera hardware and data analysis software has improved the speed, efficiency and utility of these devices.[16a,34]
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one or zero photons. The first point in the correlation function g2 is then given by
3. Recent Studies using XPCS Fluid dynamics near surfaces have been of great interest not only for the fundamental understanding of the underlying physics and fluid mechanics but also for the practical design of complex fluids for engineering applications. A liquid surface can often be regarded as a smooth profile decorated with dynamic height variations of capillary waves, which arise from the collective hydrodynamic motion of the entire liquid elements.[35] Capillary waves on a liquid surface can be either propagating or over-damped depending on the wave length, surface tension and viscosity. The wave frequency is therefore
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a complex quantity ω + iΓ, ω and Γ being the propagating and damping frequencies, respectively. The propagating mode is unstable on highly viscous liquids such as simple liquids at low temperatures or polymer melts, leaving only the over-damped mode. Capillary waves play important roles in many surface and interface-related phenomena, such as extra roughening of the interface between immiscible polymer blends, spinodal dewetting and droplet coalescence.[36] Detection of the structure and dynamics of these capillary waves, on the other hand, provides a unique and powerful method for determining the rheological properties, and understanding other physical properties such as glass transitions in confined geometry and liquid-substrate interactions. For instance, the capillary wave relaxation time constant of viscous liquid films is proportional to the ratio of viscosity to the surface tension η/γ, thus providing an alternative measure of thin film viscosities.[6b,37] More recently, Sikorski et al. and Chushkin et al. measured the capillary wave relaxation rates of super cooled organic glass forming liquids of dibutyl phthalate and polypropylene glycol, respectively, and discovered strong evidence of surface induced elasticity.[38] In another study, Evans et al.[39] by measuring the top layer capillary wave relaxations, reported that the stiffening of the polystyrene film depends on the modulus of the underlying substrates near the glass transition temperature (Tg + 9 °C) even for very thick films, implying a strong effect of the substrate on the film over a substantial length scale even exceeding both the cooperative length of segmental mobility and the radius of gyration. In this review, we will briefly go through some historical but noteworthy experimental studies using surface XPCS and discuss recent progress on the dynamical study of surfaces of a variety of systems including bulk liquids, liquid crystals, thin films and polymer/nanoparticle composites. As discussed in Section 2(b), one of the important advantages of XPCS compared to DLS is that by performing XPCS at the grazing-incidence geometry one obtains the surface sensitivity to distinguish the dynamics from the surface and the bulk. This is achieved by choosing appropriate incident angles, i.e., below the critical angle for total external reflection, the evanescent wave only penetrates a few nanometers into the surface and therefore only fluctuations of the surface region are probed. If the incident angle is chosen such that X-ray standing waves are generated within the film of interest, scattering from a certain depth of the film, rather than the surface, dominates the total scattering,[40] therefore providing information about dynamics in the interior of the film. (a) Surface dynamics of low viscosity liquids On the surface of a low viscosity liquid or for very small wave vectors, the energy dissipation rate due to the friction between liquid elements is usually not high enough to prevent wave propagation, i.e, ω is finite. Both propagating and damping modes are present. Their frequencies are usually as high as kHz, which requires very fast detection (usually a point detector such as an avalanche photodiode running at several MHz) and sufficient coherent flux. Therefore, the XPCS experiment is often performed in the heterodyne mode by mixing a static reference beam in order to increase the signal-to-noise ratio, however, at a cost of coherence contrast.
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Figure 5. (a) Autocorrelation function g2 of a liquid water surface at 5 °C and wave vector q|| = 3.4 × 10−6 Å−1. Solid line is the fitting to the heterodyne correlation function. (b) Measured dispersion relation of the propagating capillary waves fitted to ω = (γ q||3 / ρ )1/2 . Inset shows corresponding damping constants. Reproduced with permission.[5b] Copyright 2003, American Physical Society.
In grazing-incidence geometry, specular reflection often serves as the reference beam.[14d] The heterodyne mixing in surface XPCS was first observed by Gutt et al.[5b] from a liquid water surface, where both the first and second order correlation functions are present in the intensity-intensity auto-correlation function I(q , t )I(q , t + τ ) t ~ 2I s Ir Re [ g 1 (q , τ )] + I s2 g 2 (q , τ )
(31)
with g 1 (q , τ ) = cos(ωτ )exp( −Γτ ) and g 2 (q , τ ) = exp( −2Γτ ) (Figure 5(a)). The measured dispersion relation is well fitted to the theoretical predictions for an ideal incompress1 /2 ible liquid ω = ( γ q ||3 / ρ ) [22] (Figure 5(b)). Although the damping constant is higher than the prediction Γ = 2ηq2 / ρ2 using the bulk viscosity, it does exhibit the required q|| dependence. The dynamic scattering function S(q,ω) for a simple incompressible liquid can be obtained from linear response theory.[23] A critical wave vector, q||,c = 4γρ/5η 2, can then be deduced
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Adv. Mater. 2014, 26, 7764–7785
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have to be solved numerically and fit the experiment data very well throughout the entire q range (Figure 6(b)). Gutt et al.[44] have studied the XPCS from partially wetting liquid films on relatively smooth substrates such as silicon and glass. The liquids were n-hexane and cyclohexane introduced at vapor pressures well below saturation onto slightly cooled substrates. They found that very thin films (6 nm thickness) conformed to the substrate while thicker films (10 nm to 60 nm thickness) showed characteristic q−3 power-law dependence for the off-specular diffuse scattering, seen previously for partially wetting films. Interestingly, the coherent scattering showed no dynamics at all, as evidenced by static speckle patterns over time scales from 1s to 20 mins. They ascribed this puzzling result to the breakdown of the classical hydrodynamic theory for the capillary fluctuations on liquid films for partially wetting films, probably due to different boundary conditions.
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from S(q,ω)[41] so that beyond this wave vector the capillary waves are over-damped with Γ = 2η /γ q .[42] This is essentially the asymptotic limit of equation (18) as the film thickness approaches infinity. When q
