Diffusion and Thermal Diffusion by Means of Dynamic Light Scattering and Laser Holography

(5.1)

where n _iis a unit vector in the direction of the incident electric field, E ₀ is the amplitude, k _iis the wave vector, and ω _iis the angular frequency. The distance between adjacent maxima of waves defines the wavelength λ ₀ of light, and the magnitude of the wave vector is defined as $\left|{\mathbf{k}}_i\right|=2\pi n/{\lambda}_0$ , where n is the index of refraction of the medium. The angular frequency ω _iis related to the speed of light c as ${\omega}_i=2\pi c/{\lambda}_0=\left|{\mathbf{k}}_i\right|c/n$ .

In the medium, a local dielectric constant ε(r, t) is described by the average dielectric constant ε ₀ ( ${\varepsilon}_0={n}^2$ ) while assuming a nonabsorbing, nonconducting, and nonmagnetic medium,

$\boldsymbol{\upvarepsilon} \left(\mathbf{r},t\right)={\varepsilon}_0\mathbf{I}+\delta \boldsymbol{\upvarepsilon} \left(\mathbf{r},t\right).$

(5.2)

Here, δ ε(r, t) implies the fluctuation of the dielectric constant, and I is the unit tensor.

The component of the scattered electric field E _s(R, t) at a detector with a large distance R from the scattering volume is expressed as

${E}_{\mathrm{s}}\left(R,t\right)=-\frac{{q_{\mathrm{s}}}^2{E}_0}{4\pi R{\varepsilon}_0} \exp \left\{i\left({q}_{\mathrm{s}}R-{\omega}_it\right)\right\}{\displaystyle {\int}_Vd\mathbf{r}}\kern0.2em \exp \left\{i\left(\mathbf{q}\cdot \mathbf{R}\right)\right\}\left[{\mathbf{n}}_{\mathrm{s}}\cdot \delta \boldsymbol{\upvarepsilon} \left(\mathbf{r},t\right)\cdot {\mathbf{n}}_i\right]$

(5.3)

$=-\frac{{q_{\mathrm{s}}}^2{E}_0}{4\pi R{\varepsilon}_0} \exp \left\{i\left({q}_{\mathrm{s}}R-{\omega}_it\right)\right\}\delta {\varepsilon}_{i\mathrm{s}}\left(\mathbf{r},t\right),$

(5.4)

where $\delta {\varepsilon}_{i\mathrm{s}}\left(\mathbf{q},t\right)={\mathbf{n}}_{\mathrm{s}}\cdot \delta \boldsymbol{\upvarepsilon} \left(\mathbf{q},t\right)\cdot {\mathbf{n}}_i={\displaystyle {\int}_V\mathrm{d}\mathbf{r}\kern0.2em \exp \left\{i\left(\mathbf{q}\cdot \mathbf{R}\right)\right\}\left[{\mathbf{n}}_{\mathrm{s}}\cdot \delta \boldsymbol{\upvarepsilon} \left(\mathbf{r},t\right)\cdot {\mathbf{n}}_i\right]}$ is used with the definition of $\delta \boldsymbol{\upvarepsilon} \left(\mathbf{q},t\right)\equiv {\displaystyle {\int}_Vd\mathbf{r}\kern0.2em \exp \left\{i\left(\mathbf{q}\cdot \mathbf{R}\right)\right\}\delta \boldsymbol{\upvarepsilon} \left(\mathbf{r},t\right)}$ .The time correlation function of the scattered electric field is defined as ${G}^{(1)}\left(\tau \right)\equiv \left\langle {E_{\mathrm{s}}}^{*}\left(R,0\right){E}_{\mathrm{s}}\left(R,\tau \right)\right\rangle$ and is obtained as

${G}^{(1)}\left(\tau \right)=\frac{{q_{\mathrm{s}}}^4{I}_0}{16{\pi}^2{R}^2{\varepsilon_0}^2} \exp \left(-i{\omega}_i\tau \right)\left\langle \delta {\varepsilon_{\mathrm{i}s}}^{*}\left(\mathbf{q},0\right)\delta {\varepsilon}_{\mathrm{i}s}\left(\mathbf{q},\tau \right)\right\rangle,$

(5.5)

${I}_0={\left|{E}_0\right|}^2.$

(5.6)

Here, 〈 · · · 〉 means the average over them in time, which in turn is

$\left\langle {E}^{*}\left(R,0\right)E\left(R,t\right)\right\rangle =\underset{T\to \infty }{ \lim}\frac{1}{T}{\displaystyle {\int}_{-\frac{T}{2}}^{\frac{T}{2}}{E}^{*}\left(R,t\right)E\left(R,t+\tau \right)}\mathrm{d}t.$

(5.7)

The scattered light intensity I _s(t) at the detector with time t is the square of the electric field, ${I}_{\mathrm{s}}(t)={\left|{E}_{\mathrm{s}}\left(R,t\right)\right|}^2$ ; thus, it is described with a power spectrum

$I\left(\mathbf{q},{\omega}_{\mathrm{s},}R\right)=\frac{{q_{\mathrm{s}}}^4{I}_0}{16{\pi}^2{R}^2{\varepsilon_0}^2}\frac{1}{2\pi }{\displaystyle {\int}_{-\infty}^{\infty}\mathrm{d}t}\left\langle \delta {\varepsilon_{\mathrm{i}s}}^{*}\left(\mathbf{q},0\right)\delta {\varepsilon}_{\mathrm{i}s}\left(\mathbf{q},\tau \right)\right\rangle \exp \left\{i\left({\omega}_{\mathrm{s}}-{\omega}_i\right)t\right\}.$

(5.8)

For studies of biological materials in solvents, the polarization α is useful instead of the dielectric constant. When interactions among the molecules are assumed to be negligible, the polarization tensor α _jof the j-th component has the form

$\delta {\varepsilon}_{\mathrm{i}s}\left(\mathbf{r},t\right)={\displaystyle \sum_j{\alpha}_{j,\mathrm{i}s}(t)}\delta \left(\mathbf{r}-{\mathbf{r}}_j(t)\right).$

(5.9)

The number density of molecules ρ(r, t) is equivalent to $\sum_j\delta \left(\mathbf{r}-{\mathbf{r}}_j(t)\right)$ ; thus, the fluctuation of density $\delta \rho =\rho -\left\langle \rho \right\rangle$ holds:

$\delta \rho \left(\mathbf{r},t\right)={\displaystyle \int d\mathbf{r}\delta \rho \left(\mathbf{r},t\right) \exp \left(i\mathbf{q}\cdot \mathbf{r}\right)}.$

(5.10)

The dynamic structural factor $S\left(\mathbf{q},\tau \right)=\left\langle \delta {\rho}^{*}\left(\mathbf{q},0\right)\delta \rho \left(\mathbf{q},\tau \right)\right\rangle$ is then related to the correlation function of the scattered electric field by combining Eqs. (5.5), (5.9), and (5.10) as

${G}^{(1)}\left(\tau \right)=\frac{{q_{\mathrm{s}}}^4{I}_0}{16{\pi}^2{R}^2{\varepsilon_0}^2} \exp \left(-i{\omega}_i\tau \right)\left\langle {\alpha}_{j,\mathrm{i}s}(0){\alpha}_{k,\mathrm{i}s}\left(\tau \right)\right\rangle \left\langle \delta {\rho}^{*}\left(\mathbf{q},0\right)\delta \rho \left(\mathbf{q},\tau \right)\right\rangle,$

(5.11)

${G}^{(1)}\left(\tau \right)=A \exp \left(-i{\omega}_i\tau \right)\left\langle {\alpha}_{j,\mathrm{i}s}(0){\alpha}_{k,\mathrm{i}s}\left(\tau \right)\right\rangle S\left(\mathbf{q},\tau \right).$

(5.12)

Here, the proportionality constant is substituted as

$A=\frac{{q_{\mathrm{s}}}^4{I}_0}{16{\pi}^2{R}^2{\varepsilon_0}^2}.$

(5.13)

The Fourier transformation of S(q, τ) is

$S\left(\mathbf{q},\omega \right)=\frac{1}{2\pi }{\displaystyle {\int}_{-\infty}^{\infty }S\left(q,\tau \right)} \exp \left(-i\omega t\right)\mathrm{d}\tau,$

(5.14)

which is also called the dynamic structural factor. It is revealed that the correlation function is the inverse Fourier transform of the dynamic structural factor.

5.2.2 Photon Correlation Spectroscopy

The photon correlation spectroscopy measures the fluctuation of photocounts, which involves information of molecular motion. The scattered light intensity is the square of electric field ( ${I}_s(t)={\left|{E}_{\mathrm{s}}\left(R,t\right)\right|}^2$ ). When the scattering due to solvent molecules is assumed to be negligible and a single point detection at the detector surface with perfect coherence of scattered light is also assumed, the autocorrelation function of the intensity-intensity correlation function of the scattered light is written as $\left\langle I(t)I\left(t+\tau \right)\right\rangle =\left\langle I(0)I\left(\tau \right)\right\rangle \equiv {G}^{(2)}\left(\tau \right)$ , where the starting time of data acquisition is arbitrary. Thus, the intensity-intensity correlation function G ⁽²⁾(τ) is

${G}^{(2)}\left(\tau \right)=\left\langle {\mathbf{E}}^{*}(0)\mathbf{E}(0){\mathbf{E}}^{*}\left(\tau \right)\mathbf{E}\left(\tau \right)\right\rangle$

(5.15)

$=\left\langle {\mathbf{E}}^{*}(0)\mathbf{E}(0)\right\rangle \left\langle {\mathbf{E}}^{*}\left(\tau \right)\mathbf{E}\left(\tau \right)\right\rangle +\left\langle {\mathbf{E}}^{*}(0)\mathbf{E}\left(\tau \right)\right\rangle \left\langle {\mathbf{E}}^{*}\left(\tau \right)\mathbf{E}(0)\right\rangle$

(5.16)

$={\left|{G}^{(1)}(0)\right|}^2+{\left|{G}^{(1)}\left(\tau \right)\right|}^2$

(5.17)

$={\left|{G}^{(1)}(0)\right|}^2\left(1+{\left|{g}^{(1)}\left(\tau \right)\right|}^2\right).$

(5.18)

Here, ${g}^{(1)}\left(\tau \right)={G}^{(1)}\left(\tau \right)/{G}^{(1)}(0)$ is the normalized correlation function of the scattered electric field. The normalized correlation function of the scattered light g ⁽²⁾(τ) is expressed as

${g}^{(2)}\left(\tau \right)=\frac{\left\langle I(0)I\left(\tau \right)\right\rangle }{{\left\langle I(0)\right\rangle}^2}=\frac{G^{(2)}\left(\tau \right)}{{\left|{G}^{(1)}(0)\right|}^2}$

(5.19)

$=1+{\left|{g}^{(1)}\left(\tau \right)\right|}^2.$

(5.20)

Compared with Eq. (5.12), g ⁽¹⁾(τ) is equivalent to the dynamic structural factor S(q, τ). Equation (5.20) is known as the Siegert relation .

5.2.3 Diffusion in Dilute Solution

To understand the decay behavior of diffusive particles (Brownian particles), a dilute solution in which the particle obeys Fick’s second law of diffusion is assumed. Particles move more slowly than solvent molecules and contribute to a fluctuating field at the detector. The scattering function is related to the probability distribution G _s(R, t) of particles with displacement R of particles in time t,

$\frac{\partial }{\partial t}{G}_{\mathrm{s}}\left(\mathbf{R},t\right)=D{\nabla}^2{G}_{\mathrm{s}}\left(\mathbf{R},t\right).$

(5.21)

Here, D is the coefficient of self-diffusion of particles. The diffusing particles are expressed with the form ${F}_{\mathrm{s}}\left(\mathbf{q},t\right)=\left\langle \exp \left[i\mathbf{q}\cdot \left\{{\mathbf{r}}_i(t)-{\mathbf{r}}_i(0)\right\}\right]\right\rangle$ , which is related to the probability distribution

${G}_{\mathrm{s}}\left(\mathbf{R},t\right)=\left\langle \delta \left[\mathbf{R}-\left\{{\mathbf{r}}_i(t)-{\mathbf{r}}_i(0)\right\}\right]\right\rangle,$

(5.22)

and the spatial Fourier transform of G _s(R, t) has the form $\int {d}^3R \exp \left(i\mathbf{q}\cdot \mathbf{R}\right)\left\langle \delta \left[\mathbf{R}-\left\{{\mathbf{r}}_i(t)-{\mathbf{r}}_i(0)\right\}\right]\right\rangle$ .

Thus, the inverse Fourier transform gives

${G}_{\mathrm{s}}\left(\mathbf{R},t\right)=\frac{1}{{\left(2\pi \right)}^3}{\displaystyle \int {d}^3R \exp \left(-i\mathbf{q}\cdot \mathbf{R}\right){F}_{\mathrm{s}}\left(\mathbf{q},t\right)}.$

(5.23)

By comparison with Eq. (5.21), the following equation is obtained:

$\frac{\partial }{\partial t}{F}_s\left(\mathbf{q},t\right)=-{q}^2D{F}_s\left(\mathbf{q},t\right).$

(5.24)

The solution of the equation is

${g}^{(1)}\left(q,t\right)\propto {F}_{\mathrm{s}}\left(\mathbf{q},t\right)= \exp \left(-{q}^2Dt\right)$

(5.25)

$= \exp \left(-t/{\tau}_{\mathrm{D}}\right).$

(5.26)

Here, τ _D is the decay time ${\tau}_{\mathrm{D}}={\left({q}^2D\right)}^{-1}$ . This implies that the decay behavior of the correlation function depends on the scattering vector. That is, the plot of 1/τ _D vs q ² gives a straight line for Brownian particles under a noninteractive condition. The inverse of decay time $\varGamma \left(=1/{\tau}_{\mathrm{D}}\right)$ is called the decay rate . In practice, in the recent setup of the DLS experiment, homodyne detection of photocounting is often adopted, and the real system is not perfectly coherent owing to the scattering volume in the sample and a finite detector area. In this case, Eq. (5.20) is derived,

${g}^{(2)}\left(\tau \right)=B\left\{1+\beta {\left|{g}^{(1)}\left(\tau \right)\right|}^2\right\},$

(5.27)

where B is a baseline, and β refers to a coherency factor depending on the setup.

For a spherical particle with radius a and made of optically isotropic material, the Stokes-Einstein relation is useful for determining radius a from diffusion coefficient D:

$D=\frac{k_{\mathrm{B}}T}{6\pi \eta a},$

(5.28)

where k _Bis the Boltzmann’s constant , and η is the viscosity of the solvent. In the case of a randomly coiled polymer chain in solvent, the hydrodynamically equivalent radius R _H is obtained instead of particle radius a.

The experimental setups for carrying out dynamic light scattering are commercially available, and the recent suppliers are, for instance, ALV, Brookhaven, Horiba, Malvern, Otsuka, and Wyatt. The conventional setup for static light scattering can be utilized for photon counting experiments, where a correlator is necessary to obtain the correlation function.

5.2.4 Diffusion of Polymer at Uniform Temperature

Figure 5.1 shows the correlation function of scattered light g⁽²⁾(t) for poly(N-isopropylacrylamide) [PNIPAM] in ethanol (1.0 g/L). In an ideal case, the decay behavior of the correlation function is obtained by the fitting using the exponential function:

Fig. 5.1

Correlation function g⁽²⁾(t) of PNIPAM in ethanol (1.0 gL⁻¹) at 20.0 °C

${g}^{(1)}(t)= \exp \left(-t/{\tau}_{\mathrm{D}}\right).$

(5.29)

For samples with polydispersion of particle size, the single exponential form gives no good fitting results. Thus, the cumulant expansion is often used to obtain the average decay time and the distribution factors of decay time.

$\ln \left|{g}^{(1)}(t)\right|=-{K}_1t+\left({K}_2/2!\right){t}^2-\left({K}_3/3!\right){t}^3\cdots$

(5.30)

Here, K ₁ corresponds to the average decay rate (Γ), and the second term K ₂/2 and the third term K ₃/6 correlate respectively with the variance and the skewness of the distribution.

For a high-concentration sample of polymer solutions, the sum of two exponential functions is often used, or the stretched exponential form is also useful to obtain decay time τ _D.

${g}^{(1)}(t)= \exp \left[-{\left(t/{\tau}_{\mathrm{D}}\right)}^{\beta}\right]$

(5.31)

where β ( $0<\beta \le 1$ ) is the indicator of the broadness of distribution. In the case of gels, the power law relation gives good fitting results in the long delay time region.

${g}^{(1)}(t)={\left(1+t/\tau \prime \right)}^{-\phi }$

(5.32)

where τ ′ and ϕ are the cutoff time and the power law exponent, respectively. This means that the decay behavior is independent of the time scale, i.e., the gel cluster obtains a self-similar structure on a large scale compared with the molecular size.

The inverse Laplace transformation program (constrained regularization program, CONTIN ) developed by Provencher is also used to fit the correlation function [8, 9]

${g}^{(1)}(t)={\displaystyle \int G\left(\tau \right) \exp \left(-t/\tau \right)\mathrm{d}\tau }$

(5.33)

$={\displaystyle \int \tau G\left(\tau \right) \exp \left(-t/\tau \right)\mathrm{d}\left( \ln \tau \right)}$

(5.34)

with ${\displaystyle \int G\left(\tau \right)\mathrm{d}\tau }=1$ . This indicates that g ⁽¹⁾(t) is the Laplace transformation of the distribution function of decay time G(τ). It is necessary, in experiments and analyses, to consider the appropriate function to fit the correlation function, and preliminary information on the samples such as shape and concentration will be helpful in selecting the functions. It is also necessary to consider the usage of the CONTIN program whether or not the reliable fitting results are obtained.

Figure 5.2 shows the plot of Γ vs q ² of PNIPAM in ethanol. It is a straight line intersecting the origin. The decay rate Γ is obtained by a least-squares fitting using Eq. (5.30) for the correlation function shown in Fig. 5.1. According to Eqs. (5.25) and (5.26), the slope in Fig. 5.2 corresponds to the diffusion coefficient of PNIPAM in ethanol. The example of the distribution function of decay time G(τ) in Eq. (5.33), obtained by the CONTIN program, is shown in Fig. 5.3.

Fig. 5.2

The decay rate Γ vs q² of PNIPAM in ethanol (1.0 gL⁻¹) at 20.0 °C

Fig. 5.3

Decay time distribution function of PNIPAM in ethanol (1.0 gL⁻¹) at 20.0 °C

5.2.5 Gel Formation Process

Studies of the dynamic process of network formation (gelation) are important for revealing the structure and properties of gels, since the structure of a three-dimensional network depends on the kinetics of junction formation. Fibrin gel is composed of associated fibrins and is the primary structural component of blood clots. Fibrinogen is a rod-shaped protein with a molecular weight of 3.4 × 10⁵ and plays essential roles in various pathological processes, including hemostasis, thrombosis, and adhesion and aggregation of platelets.

The characteristics of the sol-to-gel transition obtained so far by the light scattering method can be summarized as follows: (1) power law behavior in g ⁽²⁾(t) at the limit of decrease in β and increase in τ of the stretched exponential formula in Eq. (5.31), (2) appearance of a long-time tail in G(τ), (3) beginning of the decrease in the coherence factor in g ⁽²⁾ (t) (nonergodicity), and (4) rapid increase in scattered light intensity and its oscillating behavior. (1) to (3) are related to dynamic light scattering and indicate the utility of the dynamic light scattering method in the study of sol-to-gel transition phenomenon [10].

Figure 5.4 shows typical double logarithmic plots of the correlation function g ⁽²⁾(t) − 1 as a function of delay time t for 2.2 g/L fibrinogen in a buffer solution [11]. The gelation process is initiated by the addition of thrombin that cleaves peptide fragments on fibrinogen. The numbers in the figure indicate the elapsed time t _e (min) of the gel formation process after the addition of thrombin. The existence of a characteristic decay time less than 220 min means that the relaxation time of a growing fibrin cluster is finite, i.e., the system is in the sol state. However, g ⁽²⁾(t) begins to show a long-time tail in the range of long delay time with a gradual decrease in β in Eq. (5.31) at elapsed time t _e > 260 min. For t _e < 260 min, the stretched exponential formula shows good agreement with the long-time tail of the correlation function. In contrast, for t _e ≥ 286 min, the power law relation can be fitted well in the long delay time region. This means that the decay at t _e ≥ 286 min became independent of the time scale, i.e., the clusters composed of fibrin fibers attain a self-similar structure on a large scale compared with the molecular size. The behavior at t _e ≥ 286 min indicates that the formation of the fibrin gel is completed at this time. The initial amplitude of g ⁽²⁾(t), the coherence factor, is an indicator of the ergodic nature of the sample. When some frozen inhomogeneity exists, as a result of cluster formation, on a very large scale, this amplitude should decrease markedly.

Fig. 5.4

Double logarithmic plots of the correlation function g⁽²⁾(t) − 1 as a function of the delay time t for fibrinogen solution measured at θ = 30° with the wavelength of incident laser operated at λ = 488 nm. The numbers in the figure refer to the elapsed time of gelation process in minutes (Reprinted with permission from Kita et al. [11]. Copyright 2002 American Chemical Society)

The decay time distribution function τG(τ) for the gelling process of 2.2 g/L fibrinogen in a buffer solution is shown in Fig. 5.5 [11]. The peak of τG(τ) of the fast mode varies from the fastest relaxation (3 × 10⁻¹ to 10⁰ ms) with a sharp single peak to another slower relaxation peak (10⁰-10¹ ms). The protofibril formation from the fibrin monomer and/or oligomer is clearly identified by the appearance of crossover between the two modes. The behaviors of τG(τ) for those initial time regions of t _e < 220 min correspond to the molecular weight of the polymer becoming greater with time; that is, fibrin oligomers become larger, and the formation of protofibrils proceeds. Therefore, substantial entanglements among protofibrils are generated at an elapsed time of t _e = 220 min, and the cooperative diffusional motion due to entanglements appears. At t _e = 237 min, another peak appears at τ = 3 × 10²-10³. These results indicate that the lateral aggregation of protofibrils occurs after sufficient protofibril formation. The peak at the longest relaxation time shifts toward a slower relaxation time for 237 ≤ t _e ≤ 286 min. This suggests that lateral aggregation to form fibers occurs and that the network formation proceeds by fiber growth due to the lateral aggregation of protofibrils. At t _e = 286 min, τG(τ) showed the slowest relaxation mode, and the time corresponded to the gelation time determined by the crossover behavior of g ⁽²⁾(t). The slowest peak of τG(τ) should originate from the gelling clusters.

Fig. 5.5

5.3 Diffusion Under Temperature Gradient

5.3.1 Thermal Diffusion

In this section, we consider the diffusion process under a nonuniform temperature condition over the system where the mixture is assumed to be confined to closed reservoirs. For simplicity, a binary mixture with a stable temperature gradient is considered. Here, the pressure of the system is assumed to be uniform and convection is neglected. When a stable temperature gradient is applied into the fluid mixture, a concentration gradient arises. This is called thermal diffusion, also referred as the Ludwig-Soret effect, and thermodiffusion [7]. According to the phenomenological treatment of irreversible thermodynamics, the local entropy production σ is related to the heat flow J _q′ and the flux J ₁ of one of the components in the binary mixture:

$\sigma =-{\boldsymbol{J}}_q^{\prime}\frac{\nabla T}{T^2}-{\boldsymbol{J}}_1\frac{\nabla {\mu}_1}{c_2T}.$

(5.35)

Here, μ ₁ is the chemical potential of component 1. The concentration c ₁ of component 1 is related to the concentration of component 2 c ₂ as ${c}_1+{c}_2=1$ . Owing to the state of minimum entropy production of the system, using Onsager reciprocal relations, we obtain useful expressions on the thermal diffusion for the system. Here, the gradient of the chemical potential is expressed with the concentration gradient

$\nabla {\mu}_1={\left(\frac{\partial {\mu}_1}{\partial {c}_1}\right)}_{p,T,{c}_2}\nabla {c}_1.$

(5.36)

The phenomenological equations for the heat flow J _q′ and the flux J ₁ of component 1 are J _q^′

$=-{L}_{qq}\frac{\nabla T}{T^2}-{L}_{q1}\frac{{\left(\partial {\mu}_1/\partial {c}_1\right)}_{p,T}}{\left(1-{c}_1\right)T}\nabla {c}_1,$

(5.37)

${\boldsymbol{J}}_1=-{L}_{1q}\frac{\nabla T}{T^2}-{L}_{11}\frac{{\left(\partial {\mu}_1/\partial {c}_1\right)}_{p,T}}{\left(1-{c}_1\right)T}\nabla {c}_1,$

(5.38)

where L means the phenomenological coefficients for respective forces or quantities. In practice, the useful expressions of these coefficients are the diffusion coefficient D and the thermal diffusion coefficient D _T, which are expressed as

$D=\frac{L_{11}}{\rho {c}_2T}{\left(\frac{\partial {\mu}_1}{\partial {c}_1}\right)}_{p,T},$

(5.39)

${D}_{\mathrm{T}}=\frac{L_{1q}}{\rho {c}_1{c}_2{T}^2}.$

(5.40)

The equation for the transport of component 1 is recalled:

${\boldsymbol{J}}_1=-\rho D\nabla {c}_1-\rho {c}_1{c}_2{D}_T\nabla T.$

(5.41)

At the steady state, the concentration gradient is developed in the binary mixture of interest, where the flux of components apparently vanishes ( ${\boldsymbol{J}}_1=0$ ). This leads to the expression of the magnitude of the concentration gradient generated by the temperature gradient as

$\nabla {c}_1=-\frac{c_1{c}_2{D}_T}{D}\nabla T.$

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