Experimental Methods

Abstract

Throughout this book we compared the predictions of theoretical models with available experimental data. This chapter is aimed at providing a reader with a flavor of experimental methods used in nucleation research. As in the previous chapters, we focus on vapor to liquid nucleation as most of the experimental studies refer to this type of transition.

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Throughout this book we compared the predictions of theoretical models with available experimental data. This chapter is aimed at providing a reader with a flavor of experimental methods used in nucleation research. As in the previous chapters, we focus on vapor to liquid nucleation as most of the experimental studies refer to this type of transition.

Prior to 1960–1970s most experiments dealt with the critical supersaturation measurements, or more generally, the conditions accompanying the onset of nucleation at various temperatures (for review see [1]). This research was pioneered by Wilson in the end of the 19th century [2] who studied the behavior of water vapor in expansion chamber and observed the onset of the condensation process and the associated with it light scattering. The main conclusion drawn from Wilson’s experiments is that if the vapor is sufficiently supersaturated, thermal density fluctuations trigger droplet formation in the chamber in the absence of impurities—the process we now refer to as homogeneous nucleation. Starting with 1970s a number of newly developed techniques appeared which make it possible to measure not only the onset conditions but the nucleation rates themselves at various temperatures and supersaturations. This big step in experimental research opened the way for quantitative tests of nucleation theories (within the accessible range of temperatures and pressures). At present quantitative nucleation rate measurements, using various experimental techniques, span the range of nucleation rates from $10^{-3}\,\,\mathrm{cm }^{-3}\,\mathrm s ^{-1}$ to $10^{18}\,\,\mathrm{cm ^{-3}}\,\mathrm s ^{-1}$. Also combination of these measurements with nucleation theorems, studied in Chap. 4, provides direct information on the properties of critical cluster.

Among the variety of methods (for a review see e.g. [3]) we describe the four most widely used techniques:

• thermal diffusion cloud chamber

• expansion cloud chamber

• shock tube

• supersonic nozzle

16.1 Thermal Diffusion Cloud Chamber

The thermal diffusion cloud chamber consists of two metallic cylindrical plates separated by the optically transparent cylindrical ring. The region between the plates forms the working volume of the chamber. The substance under study is placed as a shallow liquid pool on the lower plate of the chamber and the working volume is filled by the carrier gas (aiming at removal of the latent heat emerging in the process of condensation). The lower plate is heated while the upper plate is cooled. Due to the temperature difference $\varDelta T$ between the plates, vapor¹ evaporates from the top surface of the liquid pool, diffuses through a noncondensable carrier gas (usually helium, argon or nitrogen), and condenses on the lower surface of the top plate. Construction of the diffusion cloud chamber is illustrated in Fig. 16.1.

Fig. 16.1

Cutaway view of diffusion cloud chamber. (Reprinted with permission from Ref. [4], copyright (1975), American Institute of Physics.)

Thermal diffusion gives rise to the profiles of temperature, density, pressure and supersaturation inside the chamber. These profiles can be calculated from the one-dimensional energy and mass transport equations using an appropriate equation of state for the vapor/carrier gas mixture as shown in Fig.  16.2. At certain values of $\varDelta T$ the supersaturation in the chamber becomes sufficiently large to cause nucleation of droplets which are subsequently detected by light scattering using a laser and a photo-multiplier.

Fig. 16.2

Profiles of density, temperature, supersaturation and the nucleation rate inside the chamber. (Reprinted with permission from Ref. [5], copyright (1989), American Institute of Physics.)

Diffusion cloud chamber can operate in the temperature range from near the triple point (of the substance under study) up to the critical temperature and in the pressure range from below the ambient to elevated pressures. A typical range of accessible nucleation rates is $10^{-3}-10^{3}\,\,\mathrm{cm^{-3}\,s^{-1} }$. The growing droplets are removed by gravitational sedimentation or by convective flow which ensures the steady-state self-cleaning operational conditions. Due to this feature and to the relatively low nucleation rates, e.g. relatively small number of droplets to be counted, the quantitative nucleation rate measurements are straightforward [4–8]. Measuring nucleation rate as a function of supersaturation at a constant temperature, one can determine the size of the critical cluster using the nucleation theorem (Chap. 4). The experimentally determined critical cluster can then be compared to the nucleation models.

Note, however, that nonlinear temperature and pressure profiles inside the chamber can lead to substantial nonuniformities of temperatures and supersaturations in the working volume making it difficult to assign particular values to supersaturations and temperatures corresponding to the observed nucleation rates.

16.2 Expansion Cloud Chamber

Compared to the diffusion chamber, functioning of the expansion cloud chamber relies upon a different mechanism: rapid adiabatic expansion of the vapor/gas mixture which produces supersaturation and subsequent nucleation. The device can be generally described as a cylinder piston-like structure containing the vapor/gas mixture in the region above the cylinder and bounded by the piston walls [9] (or valves connecting additional volumes to the chamber, as in the nucleation pulse chamber of Ref. [10]). Initially the mixture has the temperature of the piston wall and the vapor may or may not be saturated. After rapid withdrawal of the piston adiabatic cooling occurs: the pressure and temperature of the mixture decrease. As a result, the supersaturation of the vapor

$$\begin{aligned} S = \frac{p^\mathrm v }{p_\mathrm{sat }(T)} \end{aligned}$$

increases because the decrease of its partial pressure $p^\mathrm v$ during the isentropic expansion is slower than the exponential decrease of the saturation pressure $p_\mathrm{sat }$ with temperature, which is given by the Clapeyron Eq. (2.14). As opposed to the diffusion chamber, nucleation of the supersaturated vapor in the working volume of the expansion chamber takes place at uniform conditions.

Among various modifications of the expansion camber—single-piston chamber [11, 12], piston-expansion tube [13]—we will describe in a somewhat more detail the nucleation pulse chamber (NPC) [10, 14, 15]. In order to ensure the constant conditions during the nucleation period a small recompression pulse is issued in NPC after the completion of the adiabatic expansion, which terminates the nucleation process after a short time, of the order of 1 ms, called the nucleation pulse. Condensational growth of droplets results in further reduction of the supersaturation due to vapor depletion. Schematically evolution of the supersaturation in the NPC during the nucleation experiment is depicted in Fig. 16.3. The cooling rate in the NPC is of the order $10^4$ K/s.

Fig. 16.3

Time dependence of the supersaturation in the expansion cloud chamber during a single nucleation experiment. Experiment starts when vapor is supersaturated. As a result of adiabatic expansion vapor becomes supersaturated and nucleation occurs during the time of nucleation pulse. After that a slight recompression terminates nucleation process; condensational growth of droplets results in further reduction of the supersaturation due to vapor depletion. (Reprinted with permission from Ref. [10], copyright (1994), American Chemical Society.)

The values of supersaturation and temperature corresponding to the measured nucleation rate are calculated from the following considerations. If $p_0$ is the initial total pressure of the vapor/gas mixture, $T_0$ and $y$ is the vapor molar fraction, then

$$\begin{aligned} p_0^\mathrm{v } = y\,p_0 \end{aligned}$$

is the partial vapor pressure at the initial conditions. After adiabatic expansion the total pressure drops by $\varDelta p_\mathrm{expt }$ becoming equal to

$$\begin{aligned} p=p_0-\varDelta p_\mathrm{expt } \end{aligned}$$

The nucleation temperature follows the Poisson law:

$$\begin{aligned} \frac{T}{T_0} = \left(\frac{p}{p_0}\right)^{(\kappa -1)/\kappa } \end{aligned}$$

(16.1)

where $\kappa = c_p/c_v$ is the ratio of specific heats for the vapor/gas mixture. Then, from the saturation vapor pressure at temperature $T$, $p_\mathrm{sat }(T)$, given by the Clapeyron Eq. (2.14), the supersaturation is found to be

$$\begin{aligned} S_{\mathrm{expt }} = \frac{y\,(p_0-\varDelta p_\mathrm{expt })}{p_\mathrm{sat }(T)} \end{aligned}$$

(16.2)

During the short (${\sim }1$ ms) nucleation pulse only a negligible fraction of the vapor is consumed, i.e. depletion effects are negligible which leaves the supersaturation practically constant. After recompression supersaturation drops, nucleation is suppressed so that only particle growth at constant number density occurs (while no new droplets are formed). Thus, the nucleation pulse method realized in the expansion tube makes it possible to decouple nucleation and growth processes.

The last step is to determine the number density $\rho _d$ of droplets formed during the nucleation pulse. In the NPC the nucleated droplets grow to the sizes ${\sim }1\,\mu$m when they are detected by the constant angle Mie scattering (CAMS) technique leading to determination of $\rho _d$. CAMS, which uses a laser operating in the visual, is based on the Mie theory of scattering of electromagnetic waves by dielectric spherical particles [16].

The basic idea behind the technique is straightforward: (i) analyzing the time evolution of the intensity of light scattered by the droplets and comparing it with the Mie theory, one finds the size $r_d$ of the droplet at time $t$; (ii) analyzing the evolution of the intensity of the transmitted light one determines the number density of droplets using the value of extinction coefficient corresponding to the droplet size $r_d$.

16.2.1 Mie Theory

To clarify CAMS, we briefly formulate the main results of the Mie theory (for details the reader is referred to Refs. [16, 17]) relevant for the analysis of nucleation experiments. Consider a single dielectric spherical particle of radius $r_d$ emerged in the vacuum and having the refractive index$m$. The particle is illuminated by the incident light with a wavelength $\lambda$. Let us introduce the dimensionless droplet radius²

$$\begin{aligned} \alpha = \left(\frac{2\pi }{\lambda }\right)\, r_d = k\,r_d \end{aligned}$$

If $I_0$ is the intensity of the incident light (watt/m$^2$), the sphere will intercept $Q_\mathrm{ext }\,\pi r_d^2\,I_0$ watt from the incident beam, independently of the state of polarization of the latter. The dimensionless quantity $Q_\mathrm{ext }(m,\alpha )$ is called the extinction efficiency. In the Mie theory it is given by

$$\begin{aligned} Q_\mathrm{ext }(m,\alpha ) = \frac{2}{\alpha ^2}\,\sum _{n=1}^\infty \,(2n+1)\,\mathfrak R (a_n+b_n) \end{aligned}$$

(16.3)

Here the complex Mie coefficients $a_n$ and $b_n$ are obtained from matching the boundary conditions at the surface of the spherical droplet. They are expressed in terms of spherical Bessel functions evaluated at $\alpha$ and $y=m\alpha$:

$$\begin{aligned} a_n&= \frac{\psi _n^{\prime }(y)\,\psi _n(\alpha )-m\, \psi _n(y)\,\psi _n^{\prime }(\alpha )}{\psi _n^{\prime }(y)\,\zeta _n(\alpha )-m\, \psi _n(y)\,\zeta _n^{\prime }(\alpha )} \nonumber \\ b_n&= \frac{m\,\psi _n^{\prime }(y)\,\psi _n(\alpha )- \psi _n(y)\,\psi _n^{\prime }(\alpha )}{m\,\psi _n^{\prime }(y)\,\zeta _n(\alpha )- \psi _n(y)\,\zeta _n^{\prime }(\alpha )} \nonumber \end{aligned}$$

where

$$\begin{aligned} \psi _n(z) =(\pi z/2)^{1/2}\,J_{n+1/2}(z) \nonumber \\ \zeta _n(z) = (\pi z/2)^{1/2}\,H^{(2)}_{n+1/2}(z) \nonumber \end{aligned}$$

and $J_{n+1/2}(z)$ is the half-integer-order Bessel function of the first kind, $H^{(2)}_{n+1/2}(z)$ is the half-integer-order Hankel function of the second kind [18].

The intensity of the incident beam decreases with the a distance $L$ (called the optical path) as it proceeds through the cloud of droplets. The transmitted light intensity is given by the Lambert-Beer law [19]

$$\begin{aligned} I_\mathrm{trans } = I_0 \mathrm{e }^{-\beta _\mathrm{ext }\,L} \end{aligned}$$

(16.4)

where $\beta _\mathrm{ext }$ is the extinction coefficient computed from

$$\begin{aligned} \beta _\mathrm{ext } = \rho _d\,\pi r_d^2\,Q_\mathrm{ext }(m,\alpha ) \end{aligned}$$

(16.5)

(here we assumed that all $\rho _d$ dielectric spheres in the unit volume are identical). The behavior of the extinction efficiency$Q_\mathrm{ext }$ as a function of the size parameter $\alpha$ is illustrated in Fig.  16.4 for the two substances with the values of refractive index$m=1.33$ (water) and $m=1.55$ (silicone oil).

Fig. 16.4

Mie extinction efficiency versus size parameter $\alpha$ for water ($m=1.33$) and silicone oil ($m=1.55$). For small particles $Q_\mathrm{ext } \sim \alpha ^4$ (Rayleigh limit); for big particles $Q_\mathrm{ext }\rightarrow 2$ (limit of geometrical optics $\alpha \rightarrow \infty$). The largest value of $Q_\mathrm{ext }$ is achieved when the particle size is close to the wavelength

Consider now the light scattered by a single sphere. The direction of scattering is given by the polar angle $\theta$ and the azimuth angle $\phi$. The intensity $I_{\mathrm{scat },1}$ of the scattered light in a point located at a large distance $r$ from the center of the particle has a form

$$\begin{aligned} I_{\mathrm{scat },1} = \frac{I_0}{k^2\,r^2}\,F(\theta ,\phi ;m,\alpha ) \end{aligned}$$

(16.6)

where $F$ is the dimensionless function of the direction (not of $r$). For the linearly polarized incident light

$$\begin{aligned} F = i_1\,\sin ^2 \phi +i_2\,\cos ^2 \phi \end{aligned}$$

(16.7)

Here $i_1$ and $i_2$ refer, respectively, to the intensity of light vibrating perpendicularly and parallel to the plain through the directions of propagation of the incident and scattered beams. The quantities $i_1$ and $i_2$ are expressed in terms of the amplitude functions $S_1(m,\alpha ;\theta )$ and $S_2(m,\alpha ;\theta )$:

$$\begin{aligned} i_1 = |S_1(m,\alpha ;\theta )|^2,~~~i_2 = |S_2(m,\alpha ;\theta )|^2 \end{aligned}$$

The amplitude functions are given by

$$\begin{aligned} S_1(m,\alpha ;\theta )&= \sum _{n=1}^\infty \,\frac{2n+1}{n(n+1)}\,[a_n\,\pi _n(\cos \theta ) + b_n\,\tau _n(\cos \theta )] \end{aligned}$$

(16.8)

$$\begin{aligned} S_2(m,\alpha ;\theta )&= \sum _{n=1}^\infty \,\frac{2n+1}{n(n+1)}\,[b_n\,\pi _n(\cos \theta ) + a_n\,\tau _n(\cos \theta )] \end{aligned}$$

(16.9)

where

$$\begin{aligned} \pi _n(\cos \theta )&= \frac{1}{\sin \theta }\,P_n^{1} (\cos \theta )\end{aligned}$$

(16.10)

$$\begin{aligned} \tau _n(\cos \theta )&= \frac{\mathrm{{d}}}{\mathrm{{d}}\theta }\,P_n^{1} (\cos \theta ) \end{aligned}$$

(16.11)

and $P_n^1 (\cos \theta )$ is the associated Legendre polynomial [18].

Averaging $F$ over the azimuth angle using the identity

$$\begin{aligned} \frac{1}{2\pi }\,\int _0^{2\pi } \sin ^2 \phi \, \mathrm{{d}}\phi =\frac{1}{2\pi } \int _0^{2\pi } \cos ^2 \phi \,\mathrm{{d}}\phi =\frac{1}{2} \end{aligned}$$

we have

$$\begin{aligned} \langle F \rangle _\phi = \frac{i_1+i_2}{2} = \frac{|S_1|^2+|S_2|^2}{2} \end{aligned}$$

(16.12)

If multiple scattering can be avoided, the total intensity of light scattered in the direction $\theta$ by all spheres in the volume $V$ is

$$\begin{aligned} I_{\mathrm{scat }} = I_{\mathrm{scat },1} \,\rho _d\,V= \frac{I_0\,\rho _d\,V}{2\,k^2r^2}\,(|S_1|^2+|S_2|^2) \end{aligned}$$

(16.13)

At fixed $m$ and $\theta$ the quantity in the round brackets as a function of $\alpha$ has a distinct pattern of maxima and minima [16].

Fig. 16.5

Time dependence of the normalized scattered light intensity (left $y$-axis) and the total pressure (right $y$-axis) for CAMS (scattering angle is $15^\circ$). It is clearly seen that scattering is detected after the nucleation pulse. (Reprinted with permission from Ref. [10], copyright (1994), American Chemical Society.)

During the single nucleation experiment one measures the time dependence of the scattering intensity$I_\mathrm{scat }^\mathrm{expt }$ (see Fig.  16.5) which has a form of a sequence of maxima and minima. Assuming that on the time scale of experiment droplets, nucleated during the pulse, grow without coagulation and Ostwald ripening, one can state that their number density $\rho _d$ remains constant. That is why in order to determine $\rho _d$ it is sufficient to compare the first peak of the function $I_\mathrm{scat }^\mathrm{expt }$ with the first maximum of the scattering intensity from the Mie theory. This comparison yields the droplet radius $r_d$ at the first peak, which after substitution into (16.4) and (16.5) yields

$$\begin{aligned} \rho _d = \frac{\frac{1}{L}\,\ln (I_0/I)}{\pi \, r_d^2\,Q_\mathrm{ext }(m,\alpha )} \end{aligned}$$

(16.14)

Note, that this procedure does not provide information about the droplet growth $r_d(t)$—the latter can be obtained from the analysis of series of peaks in the scattering intensity.

16.2.2 Nucleation Rate

The nucleation rate is calculated as

$$\begin{aligned} J = \frac{\rho _d}{\varDelta t} \end{aligned}$$

(16.15)

where $\varDelta t$ is the duration of the nucleation pulse. In various versions of expansion chambers accessible range of nucleation rates is approximately $10^2-10^9 \,\,\mathrm{cm^{-3}\,s^{-1} }$ which nicely complements the range achieved in diffusion cloud chambers.

For studies of homogeneous nucleation it is important to provide the particle-free operational regime of the chamber excluding heterogeneous effects. It has to be noted that expansion chamber is not a self-cleaning device (as the previously considered static diffusion chamber) and care has to be taken to avoid significant contamination prior to nucleation experiment.

16.3 Shock Tube

Shock tube realizes the same idea of a short nucleation pulse, providing the separation in time of the nucleation and growth processes, which we discussed in Sect. 16.2. In the shock tube this is achieved by means of the shock waves. The tube consists of two sections: the driver-, or High-Pressure Section (HPS), and the driven-, or the Low Pressure Section (LPS). The two sections are separated by the diaphragm. A small amount of condensable vapor is added to the driver section. During the nucleation experiment the diaphragm is rapidly ruptured and the high-pressure vapor/gas mixture from the driver section sets up a nearly one-dimensional, unsteady flow and the shock wave traveling from the diaphragm into the driven section. At the same time the expansion wave travels back—from the diaphragm into the driver section. Cooling of the rapidly expanding gas in the driver section imposes nucleation.

Fig. 16.6

Pulse expansion wave tube set-up. (Reprinted with permission from Ref. [20], copyright (1999), American Institute of Physics.)

The construction of the shock tube for nucleation studies was proposed by Peters and Paikert [21, 22] and further developed by van Dongen and co-workers [20, 23–26]. The scheme of the experimental set-up [20] is shown in Fig. 16.6. The HPS has a length of 1.25 m, the length of the LPS is 6.42 m. The local widening in the LPS plays an important role in creating the desired profile of pressure and, accordingly, the supersaturation: after the rupture of the polyester diaphragm between HPS and LPS the initial expansion wave traveling from LPS to HPS is followed by a set of reflections of the shock wave at the widening (see Fig. 16.7). These reflections travel back into the HPS and create the pulse-shaped expansion at the end wall of the HPS. After the short pulse and a small recompression the pressure remains constant for a longer period of time during which no nucleation occurs but the already nucleated droplets are growing to macroscopic sizes to be detected by means of the scattering technique. The temperature profile follows the adiabatic Poisson law (16.1).

Similar to the nucleation-pulse chamber, discussed in Sect. 16.2, the number density of droplets, $\rho _d$, is obtained by means of a combination of the constant-angle Mie scattering and the measured intensity of transmitted light—the procedure described in Sect. 16.2.1. In the experiments of Refs. [20, 23–25] the droplet cloud in the HPS was illuminated by the Ar-ion laser with a wavelength $\lambda = 514.2$ nm. Since the observation section in the shock tube is located near the endwall of the HPS, an obvious choice of the scattering polar angle is $\theta =90^\circ$. Figure 16.8 shows the optical set-up of the device. The laser beam passes the tube through two conical windows. The transmitted light is focused by lens $L_2$ onto photodiode $D_2$. The scattered intensity is recorded by the photomultiplier $PM$.

Fig. 16.7

Profiles of pressure and temperature and the wave propagation in the pulse expansion shock tube. (Copied from Ref. [27])

Fig. 16.8

Optical set-up used for measurements of droplet size and number density of droplets in the endwall of the shock tube. (Copied from Ref. [27])

Because of the nature of the nucleation pulse method, the value of $\rho _d$ should be approximately constant in time. The steady-state nucleation rate is given by Eq. (16.15)

$$\begin{aligned} J = \frac{\rho _d}{\varDelta t} \end{aligned}$$

where $\varDelta t$ is the duration of the pulse.

As pointed out in the previous section, besides the steady-state nucleation rate one can obtain from the same experimental data the growth law of the droplets. At each moment of time during the nucleation experiment for which the measured scattered signal is at maximum, one can find the value of the droplet radius by comparison with the corresponding maximum of the theoretical scattering intensity given by the Mie theory of Sect. 16.2.1 as illustrated in Fig. 16.9. This gives the droplet growth curve $r_d(t)$.

Fig. 16.9

Theoretical and experimental scattering patterns for n-nonane droplets. From mutual correspondence of extrema the time-resolved droplet radius $r_d(t)$ is found. (Copied from Ref. [27])

The absence of moving parts in the shock-tube (as opposed to the expansion chamber) opens a possibility to study nucleation at sufficiently high nucleation pressures—up to 40 bar—and reach nucleation rates in the range of $10^8-10^{11}\,\,\mathrm{cm }^{-3}\,\mathrm{s }^{-1}$ [20, 28].

16.4 Supersonic Nozzle

The supersonic nozzle (SSN) relies upon adiabatic expansion of the vapor/gas mixture flowing through a nozzle of some sort. The most widely used type of these devices contain the Laval (converging/diverging) nozzle [29–32]. The vapor/gas mixture is undersaturated prior to and slightly after entering the nozzle region. During the flow in the nozzle the mixture becomes saturated and then supersaturated. Nucleation and growth of the droplets takes place when the flow passes the throat region of the nozzle. Rapid increase of the supersaturation results in spontaneous onset of condensation which depletes the vapor and subsequently terminates the supersaturation.

A typical nucleation pulse is very short ${\sim }10\,\mu$s, and cooling rates are very high: ${\sim }5\times 10^5$ K/s leading to characteristic nucleation rates as high as $10^{16}-10^{18}\,\,\mathrm{cm }^{-3}\,\mathrm{s }^{-1}$. The droplets formed in SSN are extremely small ${\sim }1-20$ nm; critical embryos are even smaller ${\sim }0.1$ nm, containing 10–30 mol. Clearly, droplets of this size can not be detected by optical devices operating in the visual—shorter wavelength is required. Methods used for particle characterization in SSN are small-angle neutron scattering (SANS ) and small-angle x-ray scattering (SAXS).

Fig. 16.10

Schematic diagram of the experimental set-up with supersonic nozzle and SAXS unit. (Copied from Ref. [33])

The schematic diagram of the experimental set-up with the supersonic nozzle and SAXS unit is shown in Fig. 16.10. The experiment consists of the pressure trace measurements during the expansion and the SAXS measurements. A movable pressure probe measures the pressure profile of the gas $p(x)$ along the axis of the nozzle. The condensible vapor mole fraction $y$ is determined from the mass flow measurements. Using the stagnation conditions $p_0,\,T_0$ of the vapor/gas mixture, one determines the pressure profile of the vapor along the nozzle

$$\begin{aligned} p^\mathrm{v }(x) = y\,p_0\,\left[\frac{p(x)}{p_0}\right]\,\left[1-\frac{g(x)}{g_\infty }\right] \end{aligned}$$

where $g(x)$ is the condensate mass fraction at point $x$,

$$\begin{aligned} g_\infty = \frac{\dot{m_v}}{\dot{m_v}+\dot{m_\mathrm{gas }}} \end{aligned}$$

where $\dot{m_v},\,\dot{m_\mathrm{gas }}$ are the mass flow rates of the vapor and gas, respectively. Then, the supersaturation profile is

$$\begin{aligned} S(x) = \frac{p^\mathrm{v }(x)}{p_\mathrm{sat }(T(x))} \end{aligned}$$

where $T(x)$ is the temperature at point $x$ found from the Poisson equation.

Fig. 16.11

SAXS spectrum of n-butanol at plenum temperature $T_0=50^\circ$ C and pressure $p_0=30.2$ kPa. The solid line is a fit to Gaussian distribution of spherical sizes with parameters given by (16.16)–(16.17). (Copied from Ref. [33])

Using SAXS technique, one studies elastic scattering of X-rays by a cloud of droplets. Scattering leads to interference effects and results in a pattern, which can be analyzed to provide information about the size of droplets and their number density. Let us define a scattering vector (length) according to

$$\begin{aligned} q = \frac{4\pi }{\lambda }\,\sin (\theta /2) \end{aligned}$$

where $\theta$ is the scattering angle, $\lambda$ is wavelength of the incident beam; for SAXS $\lambda \approx 1$Å. The scattering intensity$I_\mathrm{scat }$ is proportional to the number density of particles, $\rho _d$, and the form-factor$P$ of a single particle. For a spherical particle of radius $r_d$ the form-factor takes a simple form [31]

$$\begin{aligned} P(q,r_d) = \left[ \frac{4\pi \,(\sin (qr_d)-qr_d \cos (qr_d))}{q^3}\right]^2\,\rho _{SLD}^2 \end{aligned}$$

where $\rho _{SLD}$ is the contrast factor, being the difference in the scattering length density between the liquid droplet and surrounding bulk gas. Assuming Gaussian distribution of droplet sizes with the mean $\langle r_d \rangle$ and the width $\sigma$, $I_\mathrm{scat }$ can be written as

$$\begin{aligned} I_\mathrm{scat }(q) = \rho _d\,\frac{1}{\sigma \sqrt{2\pi }}\, \int \exp \left[-\,\frac{(\langle r_d\rangle -r_d)^2}{2\sigma ^2}\right]\,P(q,r_d)\,\mathrm{{d}}r_d \end{aligned}$$

Fitting the measured scattering intensity to this expression, one finds the desired quantities $\rho _d$ and $\langle r_d \rangle$.

Figure 16.11 from Ref. [33] illustrates this procedure for SSN experiments with n-butanol at plenum temperature $T_0=50^\circ$ C and pressure $p_0=30.2$ kPa. The solid line gives the fit to the Gaussian distribution with the following set of parameters

$$\begin{aligned} \langle r_d \rangle = 7\,\mathrm{nm };~~\sigma =2.19\,\mathrm{nm } \end{aligned}$$

(16.16)

and the number density

$$\begin{aligned} N \equiv \rho _d = 1.7\times 10^{12}\,\mathrm{cm }^{-3} \end{aligned}$$

(16.17)

Taking into account the pulse duration $\varDelta t\approx 10\,\mu$s, this leads to the nucleation rate

$$\begin{aligned} J \approx 1.7\times 10^{17}\,\mathrm{cm^{-3}\,s^{-1} } \end{aligned}$$

Thus, quantitative nucleation rate measurements, using various techniques discussed in this chapter, cover the range of more than 20 orders of magnitude. These measurements, in combination with nucleation theorems of Chap. 4, also provide direct information about the properties of the critical clusters which can be compared to predictions of theoretical models.