CIS-01496; No of Pages 15 Advances in Colloid and Interface Science xxx (2014) xxx–xxx
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A novel approach to the theory of homogeneous and heterogeneous nucleation Eli Ruckenstein a,⁎, Gersh O. Berim a, Ganesan Narsimhan b a b
Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, NY 14260, United States Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, IN 47907, United States
a r t i c l e
i n f o
Available online xxxx Keywords: Nucleation Kinetic approach Fokker–Plank equation First passage time
a b s t r a c t A new approach to the theory of nucleation, formulated relatively recently by Ruckenstein, Narsimhan, and Nowakowski (see Refs. [7–16]) and developed further by Ruckenstein and other colleagues, is presented. In contrast to the classical nucleation theory, which is based on calculating the free energy of formation of a cluster of the new phase as a function of its size on the basis of macroscopic thermodynamics, the proposed theory uses the kinetic theory of fluids to calculate the condensation (W+) and dissociation (W−) rates on and from the surface of the cluster, respectively. The dissociation rate of a monomer from a cluster is evaluated from the average time spent by a surface monomer in the potential well as obtained from the solution of the Fokker–Planck equation in the phase space of position and momentum for liquid-to-solid transition and the phase space of energy for vapor-to-liquid transition. The condensation rates are calculated using traditional expressions. The knowledge of those two rates allows one to calculate the size of the critical cluster from the equality W+ = W− as well as the rate of nucleation. The developed microscopic approach allows one to avoid the controversial application of classical thermodynamics to the description of nuclei which contain a few molecules. The new theory was applied to a number of cases, such as the liquid-to-solid and vapor-to-liquid phase transitions, binary nucleation, heterogeneous nucleation, nucleation on soluble particles and protein folding. The theory predicts higher nucleation rates at high saturation ratios (small critical clusters) than the classical nucleation theory for both solid-toliquid as well as vapor-to-liquid transitions. As expected, at low saturation ratios for which the size of the critical cluster is large, the results of the new theory are consistent with those of the classical one. The present approach was combined with the density functional theory to account for the density profile in the cluster. This approach was also applied to protein folding, viewed as the evolution of a cluster of native residues of spherical shape within a protein molecule, which could explain protein folding/unfolding and their dependence on temperature. © 2014 Elsevier B.V. All rights reserved.
Contents 1. 2. 3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nucleation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-to-solid phase transition . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Dissociation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Condensation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor-to-liquid phase transition . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Dissociation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Condensation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Combination with density functional theory (DFT) . . . . . . . . . . . . 4.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Homogeneous nucleation. Uniform cluster . . . . . . . . . . . 4.4.2. Homogeneous nucleation. Nonuniform cluster . . . . . . . . . . 4.4.3. Heterogeneous nucleation. Uniform cluster on a planar solid surface
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⁎ Corresponding author. Tel.: +1 716 645 1179; fax: +1 716 645 3822. E-mail address: [email protected] (E. Ruckenstein).
http://dx.doi.org/10.1016/j.cis.2014.10.011 0001-8686/© 2014 Elsevier B.V. All rights reserved.
Please cite this article as: Ruckenstein E, et al, A novel approach to the theory of homogeneous and heterogeneous nucleation, Adv Colloid Interface Sci (2014), http://dx.doi.org/10.1016/j.cis.2014.10.011
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E. Ruckenstein et al. / Advances in Colloid and Interface Science xxx (2014) xxx–xxx
4.4.4. Uniform liquid cluster on a rough solid surface 4.4.5. Nonuniform cluster on a rough solid surface . 5. Other applications of the RNN theory . . . . . . . . . . . . 5.1. Nucleation on soluble particles . . . . . . . . . . . 5.2. Binary nucleation . . . . . . . . . . . . . . . . . 5.3. Nucleation on liquid aerosols . . . . . . . . . . . . 5.4. Nucleation mechanism of protein folding . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Nucleation – the emerging of nuclei of a new phase during a first order transition – attracted interest over the past decades due to the key role which this phenomenon plays in various natural and technological processes, including the phase transitions in atmosphere, in the confined nanoscale systems, etc. There are two different approaches to nucleation which can be named as classical and kinetic. The classical nucleation theory (CNT), developed initially by Volmer and Weber [1], Farkas [2], and Becker and Döring [3] was improved later by Kuhrt [4], Lothe and Pound [5], Reiss [6], etc. In this theory the new phase appears first as small clusters of molecules which can grow or shrink depending on the state of the initial phase. The appearance and initial growth of the cluster is due exclusively to local density fluctuations. However, only after the cluster attains a critical size its growth becomes thermodynamically favorable, and the cluster will grow irreversibly thus depleting the initial (metastable) phase. A cluster of critical size is often referred to as nucleus. The free energy of formation of a critical cluster provides the height of the activation barrier to nucleation. The nucleation rate Js, defined as the number of nuclei appearing per unit time on a unit volume, is provided in CNT by the expression Js = Kexp[− W∗/kBT], where kB is the Boltzmann constant, T the temperature in degree Kelvin, K a kinetic prefactor, and W∗ the height of the nucleation barrier. In the framework of CNT, W∗ is calculated using the macroscopic concept of surface tension which is, obviously, not applicable to objects of atomic size. The alternate approach, the kinetic nucleation theory of Ruckenstein– Narsimhan–Nowakowski (RNNT), does not involve the macroscopic concept of surface tension and determines the critical nucleus from the balance between the evaporation (dissociation) and condensation rates from and on the surface of the nucleus, respectively [7–30]. It was developed first assuming uniform density in the cluster [7–9], and extended later to nonuniform clusters [10,18,30] both for homogeneous and heterogeneous nucleation using the density functional theory. RNNT was applied to various nucleation phenomena, ranging from homogeneous nucleation in liquids and gases [8,9,12–14] to heterogeneous nucleation [15,29,30], and to such a complex case as protein folding [22–27]. The main idea of RNNT is to compare the rates of evaporation (dissociation) per unit area (W− j ) of molecules from the surface of a cluster containing j molecules and condensation (W+ j ) on such a surface. At + the critical cluster size, those rates are equal, i.e. W− j = Wj and this equation provides the size of the critical cluster. For clusters of uniform density, the rate of evaporation (dissociation) was provided by the inverse of the average time spent by a molecule in the potential well, generated by its interaction with neighboring molecules, by employing a first passage time analysis [8–20]. The first passage time is the time during which a molecule located initially in the potential well will leave the well. The calculations were based on the Fokker–Planck equation which for liquid-to-solid nucleation involved the probability density function in the phase space of position and velocity, which is approximated by the backward Smoluchowski equation. For vaporto-liquid nucleation, the energy of the molecule in the potential well is weakly dependent on its position since it undergoes very few collisions with the medium. Consequently, the first passage time can
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0 0 0 0 0 0 0 0
be calculated by employing the Fokker–Planck equation for the probability density function in the energy space. The condensation rate was obtained from the stationary solution of the diffusion equation in an external field (for liquid-to-solid phase transformation) and the kinetic theory of gases (for liquid-to-vapor nucleation). In this paper, a review of RNNT is presented by emphasizing the main ideas and calculation procedures employed. In Section 2 the expression for the nucleation rate which is used in RNNT is derived. In Section 3 the theory is applied to the liquid-to-solid phase transition and in Section 4 to the vapor-to-liquid phase transition. In Section 5, more complex cases, such as nucleation on soluble particles and aerosols (Sections 5.1 and 5.3), binary nucleation (Section 5.2) and protein folding (Section 5.4) are considered. 2. Nucleation rate During the nucleation stage of a first order phase transition, the new phase appears as a manifold of clusters of various sizes. Denoting by fi the time-dependent number of clusters consisting of i (i = 1, 2, …) molecules, the flux Ii of clusters passing from the population of clusters fi − 1 to population fi is given by [31] þ
−
Ii ¼ W i−1 f i−1 −W i f i
ð1Þ
+ where W− i and Wi are the rates of evaporation (dissociation) and condensation of molecules from and on the surface of clusters belonging to population fi, respectively, both dependent on the cluster size. The change of fi as function of time is provided by the equation
df i þ − þ − ¼ − Iiþ1 −Ii ¼ W i−1 f i−1 − W i þ W i f i þ W iþ1 f iþ1 : dt
ð2Þ
Assuming a smooth dependence of fi and W± i on i, Eq. (2) can be transformed into a partial differential equation which is more convenient for analysis than Eq. (2). That transformation can be performed in various ways which provide slightly different equations. In RNNT, two such transformations are employed. The first one, suggested by Goodrich [32], consists in expanding the continuous functions f(g, t), I(g, t), and W±(g) about the midpoint i + 1/2. As a result, the following differential equation is obtained ∂f ðg; t Þ ∂Iðg; t Þ ¼− ∂t ∂g 1 ∂2 þ ∂ þ − − W ðg Þ þ W ðg Þ f ðg; t Þ− ¼ W ðg Þ−W ðg Þ f ðg; t Þ; 2 ∂g 2 ∂g ð3Þ where the continuous variable g replaces the discrete variable i [8,9,17]. Since the time scales of formation of small unstable clusters are much smaller than the time scale of formation and growth of the critical cluster, the system attains a quasi steady state after a short time.
Please cite this article as: Ruckenstein E, et al, A novel approach to the theory of homogeneous and heterogeneous nucleation, Adv Colloid Interface Sci (2014), http://dx.doi.org/10.1016/j.cis.2014.10.011
E. Ruckenstein et al. / Advances in Colloid and Interface Science xxx (2014) xxx–xxx
The steady state flux which provides the nucleation rate Js is given by equation 1 d þ − þ − W ðg Þ þ W ðg Þ f s ðg Þ þ W ðg Þ−W ðg Þ f s ðg Þ 2 dg
Js ¼ −
ð4Þ
where fs(g) is the steady state solution of Eq. (3). The functions fs(g) and the flux Js can be obtained by solving Eq. (3) for the boundary conditions fs(g) → 0 for g → ∞ (large clusters) and fs(g) = ρs for g → 1 (smallest clusters, monomers) where ρs is the number density of monomers at infinite distance from the cluster. The first boundary condition takes into account that clusters with very large sizes are absent. The second boundary condition involves the assumption that nucleation does not cause substantial depletion of the monomers and that their number density ρs remains the initial one (in calculations, the limit 1 can be replaced by zero). Finally, the rate of nucleation, Js, is obtained by introducing the solution of Eq. (3) in Eq. (4). One thus obtains Js ¼ Z
∞ 0
þ
W ð1Þρs
1 2
exp½−2wðg Þdg
Z wðg Þ ¼
þ
−
W ðg Þ−W ðg Þ dg : þ − 0 W ðg Þ þ W ðg Þ g
Z eq þ J s ¼ n1 W ð1Þ=
ν
ð7Þ
exp
1
W þ ðν Þ dν −hðνÞ−ln þ W ð1Þ
þ
W ð j−1Þ : − j¼2 W ð jÞ
exp½hðν Þ ¼ ∏
ð11Þ
ð12Þ
The function −hðν Þ−ln
W þ ðν Þ ≡ H ðνÞ W þ ð1Þ
ð13Þ
in Eq. (11) exhibits a sharp maximum for the critical cluster size ν = ν∗ and the steepest descent method allows to write for the nucleation rate eq
ð6Þ
∞
where function h(ν) is provided by equation
þ
J s ≃n1 W ð1Þ −
It should be noted that in Eq. (6), the rates of dissociation, W−(g), and condensation, W+(g), while functions of the discrete number of molecules in the cluster, are considered as continuous functions of g, or, equivalently, of the cluster radius R. The function w(g) exhibits a sharp minimum at the critical cluster size g = g∗ and the integral in Eq. (5) can be calculated using the steepest descent method. As a result, the rate of nucleation acquires the form !1=2 1 þ w″ ðg Þ J s ¼ W ð1Þρs exp½2wðg Þ 2 π
Using the stationary solution of the above equation for the boundary eq conditions, f(1)/neq 1 = 1 and f(ν)/n (ν) → 0 as ν → ∞, one obtains the steady-state nucleation rate Js
ð5Þ
where W+(1) is the condensation rate calculated for clusters consisting of one molecule and
3
!1=2 H ″ ðν Þ expð−H ðν ÞÞ: 2π
ð14Þ
Eq. (14) was used for the description of nucleation rate for vapor-toliquid transition. More details about the transformation from the discrete equation Eq. (2) to continuous differential equations are presented in Ref. [11]. For comparison, the steady-state nucleation rate of CNT for vaporto-liquid transition is provided by the equation CNT
Js
eq
¼
þ
n1 W ð1Þ ση2 kB T 4πρη3
!1=2 exp −
! 16πσ 3 3ðkB T Þ3 ρ2 ln2 ðζ Þ
ð15Þ
where η is diameter of vapor molecule, ρ the vapor density, ζ the vapor supersaturation, and σ the liquid–vapor surface tension. 3. Liquid-to-solid phase transition 3.1. Dissociation rate
where w″(g) is the second derivative of w(g) with respect to g. To determine w(g), one should find first the rates W+(g) and W−(g) that is one of the main goals of RNNT. Eqs. (5) and (7) were used in Refs. [8–10] for calculating the nucleation rate in liquid-to-solid transformation. The second type of transformation from the discrete Eq. (2) to a continuous one involves the detailed balance þ eq
−
eq
W i ni ¼ W iþ1 niþ1
ð8Þ
which provides the equilibrium solution of the differential equation obtained from Eq. (2) [33]. In Eq. (8), neq i is the equilibrium number of clusters containing i molecules. Eq. (8) can be applied recursively to obtain eq
eq
W þj−1 − j¼2 W j i
ni ¼ n1 ∏
ð9Þ
where neq 1 is the equilibrium number of clusters consisting of single molecules. Expressing the discrete cluster sizes by the continuous variable ν and combining Eq. (2) with Eq. (8), the evolution of the cluster distribution f(ν, t) can be described by (see Refs. [11,13]) ∂f ðν; t Þ ∂ ∂ f ðν; t Þ ∂ þ eq ¼ − Iðν; t Þ: ¼ W ðν Þn ðν Þ eq ∂t ∂ν ∂ν n ðν Þ ∂ν
ð10Þ
Let us consider a spherical cluster of a solid phase surrounded by a liquid. The molecules of the cluster are held together by attractive forces. Because of collisions with the molecules of the surrounding medium, a surface monomer exhibits Brownian motion in the attractive potential well and dissociates from the cluster as soon as it surpasses the range of the interaction potential. The rate of dissociation of surface monomers is provided by the kinetic law of a first order reaction W
−
¼
Ns bτN
ð16Þ
where Ns is the number of surface monomers and b τ N is the mean first passage time, characterizing the average time required for a surface monomer to cross the outer boundary of the potential well. The Brownian motion of a molecule in the potential well is described by a Fokker–Plank equation for the joint probability density function f(r, v, t) in the phase space of position r and velocity v as [7] ∂f ∇Φ k T ∇v f ¼ γ f ∇v v f þ B ∇v f ; þ v ∇r f − m m ∂t
ð17Þ
where Φ(r) is the interaction potential between a molecule of liquid and those of the cluster, m is the mass of a surface monomer, γf is the friction coefficient with the surrounding medium. Because in liquids the relaxation time to achieve equilibrium velocity distribution (which is of the order of γ−1 f ) is much shorter than the dissociation time, the
Please cite this article as: Ruckenstein E, et al, A novel approach to the theory of homogeneous and heterogeneous nucleation, Adv Colloid Interface Sci (2014), http://dx.doi.org/10.1016/j.cis.2014.10.011
4
E. Ruckenstein et al. / Advances in Colloid and Interface Science xxx (2014) xxx–xxx
Fokker–Plank equation reduces to the Smoluchowski equation, which describes diffusion in an external field [34,35]. In the case of spherical symmetry, this equation has the form ∂nðr; t Þ −2 ∂ 2 −Φðr Þ ∂ Φðr Þ ¼ Dr r e e nðr; t Þ ∂t ∂r ∂r
ð18Þ
where n(r, t) is the number density of diffusing surface monomers in their potential wells at position r and time t, D the diffusion coefficient, r the radial distance from the center of symmetry, and Φðr Þ ¼ Φðr Þ=kB T. To determine b τ N, the first passage time τ(r0) for a molecule that is initially located at the distance r0 from the center of the cluster must be first calculated and then averaged by integrating over r0 for the initial distribution n(r0, 0) of molecules. For calculations, it is more convenient to deal with the backward evolution of the conditional probability density w(r, t|r0, 0) which provides the probability for the monomer to be located at the final (fixed) position r at time t given that its initial position is r0. The corresponding backward Fokker–Planck (backward Smoluchowski) equation is given by ∂ −2 Φðr Þ ∂ 2 −Φðr 0 Þ ∂ r0 e wðr; tjr 0 ; 0Þ : wðr; tjr 0 ; 0Þ ¼ Dr0 e 0 ∂t ∂r 0 ∂r 0
ð19Þ
Z
Rþη R−η
wðr; tjr 0 ; 0Þdr :
τðR þ ηÞ ¼ 0
whereas the boundary condition Eq. (23) can be rewritten as a reflecting boundary condition
dτ
¼0 : dr ðR−ηÞ
The solution of Eq. (25) for the above boundary conditions yields the following expression for τ(r0) τðr 0 Þ ¼
The above expression involves the assumption that the interaction potential is symmetric around the potential well located at R. Integrating Eq. (19) with respect to r one obtains, ð21Þ
If the initial position of the monomer is (R + η), it will dissociate instantaneously and consequently [7,35], ð22Þ
for all t. The radial distance of the surface monomer cannot be less than (R − η) because of the hard sphere repulsion with the neighboring monomers. Consequently, the flux of the monomer at (R − η) is zero [7,35], i.e. ∂ Q ðr; t Þjr¼R−η ¼ 0 : ∂r
ð23Þ
From the definition of Q(r0, t), one can see that the probability for the dissociation time of a monomer to be between 0 and t is 1 − Q(r0, t). The first passage time τ(r0) is therefore given by [7] Z τðr 0 Þ ¼ −
∞ 0
t
∂Q ðr0 ; t Þ dt ¼ ∂t
Z
∞ 0
Q ðr 0 ; t Þdt:
ð24Þ
Integrating Eq. (21) with respect to t and taking into account that Q(r0, ∞) = 0, one obtains −2 Φðr Þ ∂ 2 −Φðr 0 Þ ∂ r0 e τðr 0 Þ ¼ 1 : −Dr 0 e 0 ∂r 0 ∂r 0
1 D
Z
Rþη
x
−2 ΦðxÞ
ð25Þ
Z
x
2 −ΦðxÞ
e
y e ðR−ηÞ
0
dydx:
ð28Þ
Therefore, the average first passage time b τ N for all surface monomers at any initial location can be obtained by averaging Eq. (28) with respect to the equilibrium distribution of surface monomers within the potential well given by, =Z
ð29Þ
where Z is the partition function Z Z¼
Rþη
2 −ΦðxÞ
x e R−η
dx :
ð30Þ
One thus obtains Z bτN ¼
Q ðR þ η; t Þ ¼ 0
ð27Þ
2 −ΦðxÞ
ð20Þ
∂ −2 Φðr Þ ∂ 2 −Φðr0 Þ ∂ r0 e Q ðr 0 ; t Þ : Q ðr 0 ; t Þ ¼ Dr 0 e 0 ∂t ∂r 0 ∂r 0
ð26Þ
peq ðxÞ ¼ x e
A surface monomer is associated with the cluster as long as its radial position is less than (R + η), η being the range of the interaction potential. The probability Q(r0, t) for the dissociation time of the surface monomer to be greater than t provided that its initial position is r0, is given by Q ðr0 ; t Þ ¼
The boundary condition Eq. (22) can be rewritten as an absorbing boundary condition
Rþη R−η
τðxÞpeq ðxÞdx :
ð31Þ
3.2. Condensation rate The condensation rate W+ is obtained from the stationary solution of the diffusion equation in an external field and has the form þ
W ¼ γ4πDRρs
ð32Þ ∞
−2 Φðr Þ
where γ−1 ¼ R∫Rþη r e dr. Here and below, the variable g (number of molecules in the cluster) in W±(g) is omitted. The relation between g and R is provided by the expression g ¼ 4π3 ρc R3 where ρc is the number density of molecules in the cluster. The rate of condensation is affected (through γ) by the potential field Φ(r). Note that in the limit R → ∞, the factor γ tends to unity and Eq. (32) leads to an expression for the condensation rate in the absence of an external field [9]. The size Rc of the critical nucleus can be calculated by using the equality between the dissociation and condensation rates (W+ = W−) provided by Eqs. (16) and (32). 3.3. Results Applications of RNNT to liquid-to-solid nucleation were carried out for amorphous spherical clusters (Refs. [8,9,28]), and fcc and icosahedral clusters (Ref. [10]). In general, the cluster (with number density ρc) is surrounded by molecules of solvent (number density ρw) and molecules of solute (number density ρs). The interaction potential between any two molecules was selected as a combination of a hard core repulsion and dispersion attraction ϕðr 12 Þ
¼
8
Contents lists available at ScienceDirect
Advances in Colloid and Interface Science journal homepage: www.elsevier.com/locate/cis
A novel approach to the theory of homogeneous and heterogeneous nucleation Eli Ruckenstein a,⁎, Gersh O. Berim a, Ganesan Narsimhan b a b
Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, NY 14260, United States Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, IN 47907, United States
a r t i c l e
i n f o
Available online xxxx Keywords: Nucleation Kinetic approach Fokker–Plank equation First passage time
a b s t r a c t A new approach to the theory of nucleation, formulated relatively recently by Ruckenstein, Narsimhan, and Nowakowski (see Refs. [7–16]) and developed further by Ruckenstein and other colleagues, is presented. In contrast to the classical nucleation theory, which is based on calculating the free energy of formation of a cluster of the new phase as a function of its size on the basis of macroscopic thermodynamics, the proposed theory uses the kinetic theory of fluids to calculate the condensation (W+) and dissociation (W−) rates on and from the surface of the cluster, respectively. The dissociation rate of a monomer from a cluster is evaluated from the average time spent by a surface monomer in the potential well as obtained from the solution of the Fokker–Planck equation in the phase space of position and momentum for liquid-to-solid transition and the phase space of energy for vapor-to-liquid transition. The condensation rates are calculated using traditional expressions. The knowledge of those two rates allows one to calculate the size of the critical cluster from the equality W+ = W− as well as the rate of nucleation. The developed microscopic approach allows one to avoid the controversial application of classical thermodynamics to the description of nuclei which contain a few molecules. The new theory was applied to a number of cases, such as the liquid-to-solid and vapor-to-liquid phase transitions, binary nucleation, heterogeneous nucleation, nucleation on soluble particles and protein folding. The theory predicts higher nucleation rates at high saturation ratios (small critical clusters) than the classical nucleation theory for both solid-toliquid as well as vapor-to-liquid transitions. As expected, at low saturation ratios for which the size of the critical cluster is large, the results of the new theory are consistent with those of the classical one. The present approach was combined with the density functional theory to account for the density profile in the cluster. This approach was also applied to protein folding, viewed as the evolution of a cluster of native residues of spherical shape within a protein molecule, which could explain protein folding/unfolding and their dependence on temperature. © 2014 Elsevier B.V. All rights reserved.
Contents 1. 2. 3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nucleation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-to-solid phase transition . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Dissociation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Condensation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor-to-liquid phase transition . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Dissociation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Condensation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Combination with density functional theory (DFT) . . . . . . . . . . . . 4.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Homogeneous nucleation. Uniform cluster . . . . . . . . . . . 4.4.2. Homogeneous nucleation. Nonuniform cluster . . . . . . . . . . 4.4.3. Heterogeneous nucleation. Uniform cluster on a planar solid surface
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⁎ Corresponding author. Tel.: +1 716 645 1179; fax: +1 716 645 3822. E-mail address: [email protected] (E. Ruckenstein).
http://dx.doi.org/10.1016/j.cis.2014.10.011 0001-8686/© 2014 Elsevier B.V. All rights reserved.
Please cite this article as: Ruckenstein E, et al, A novel approach to the theory of homogeneous and heterogeneous nucleation, Adv Colloid Interface Sci (2014), http://dx.doi.org/10.1016/j.cis.2014.10.011
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E. Ruckenstein et al. / Advances in Colloid and Interface Science xxx (2014) xxx–xxx
4.4.4. Uniform liquid cluster on a rough solid surface 4.4.5. Nonuniform cluster on a rough solid surface . 5. Other applications of the RNN theory . . . . . . . . . . . . 5.1. Nucleation on soluble particles . . . . . . . . . . . 5.2. Binary nucleation . . . . . . . . . . . . . . . . . 5.3. Nucleation on liquid aerosols . . . . . . . . . . . . 5.4. Nucleation mechanism of protein folding . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Nucleation – the emerging of nuclei of a new phase during a first order transition – attracted interest over the past decades due to the key role which this phenomenon plays in various natural and technological processes, including the phase transitions in atmosphere, in the confined nanoscale systems, etc. There are two different approaches to nucleation which can be named as classical and kinetic. The classical nucleation theory (CNT), developed initially by Volmer and Weber [1], Farkas [2], and Becker and Döring [3] was improved later by Kuhrt [4], Lothe and Pound [5], Reiss [6], etc. In this theory the new phase appears first as small clusters of molecules which can grow or shrink depending on the state of the initial phase. The appearance and initial growth of the cluster is due exclusively to local density fluctuations. However, only after the cluster attains a critical size its growth becomes thermodynamically favorable, and the cluster will grow irreversibly thus depleting the initial (metastable) phase. A cluster of critical size is often referred to as nucleus. The free energy of formation of a critical cluster provides the height of the activation barrier to nucleation. The nucleation rate Js, defined as the number of nuclei appearing per unit time on a unit volume, is provided in CNT by the expression Js = Kexp[− W∗/kBT], where kB is the Boltzmann constant, T the temperature in degree Kelvin, K a kinetic prefactor, and W∗ the height of the nucleation barrier. In the framework of CNT, W∗ is calculated using the macroscopic concept of surface tension which is, obviously, not applicable to objects of atomic size. The alternate approach, the kinetic nucleation theory of Ruckenstein– Narsimhan–Nowakowski (RNNT), does not involve the macroscopic concept of surface tension and determines the critical nucleus from the balance between the evaporation (dissociation) and condensation rates from and on the surface of the nucleus, respectively [7–30]. It was developed first assuming uniform density in the cluster [7–9], and extended later to nonuniform clusters [10,18,30] both for homogeneous and heterogeneous nucleation using the density functional theory. RNNT was applied to various nucleation phenomena, ranging from homogeneous nucleation in liquids and gases [8,9,12–14] to heterogeneous nucleation [15,29,30], and to such a complex case as protein folding [22–27]. The main idea of RNNT is to compare the rates of evaporation (dissociation) per unit area (W− j ) of molecules from the surface of a cluster containing j molecules and condensation (W+ j ) on such a surface. At + the critical cluster size, those rates are equal, i.e. W− j = Wj and this equation provides the size of the critical cluster. For clusters of uniform density, the rate of evaporation (dissociation) was provided by the inverse of the average time spent by a molecule in the potential well, generated by its interaction with neighboring molecules, by employing a first passage time analysis [8–20]. The first passage time is the time during which a molecule located initially in the potential well will leave the well. The calculations were based on the Fokker–Planck equation which for liquid-to-solid nucleation involved the probability density function in the phase space of position and velocity, which is approximated by the backward Smoluchowski equation. For vaporto-liquid nucleation, the energy of the molecule in the potential well is weakly dependent on its position since it undergoes very few collisions with the medium. Consequently, the first passage time can
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be calculated by employing the Fokker–Planck equation for the probability density function in the energy space. The condensation rate was obtained from the stationary solution of the diffusion equation in an external field (for liquid-to-solid phase transformation) and the kinetic theory of gases (for liquid-to-vapor nucleation). In this paper, a review of RNNT is presented by emphasizing the main ideas and calculation procedures employed. In Section 2 the expression for the nucleation rate which is used in RNNT is derived. In Section 3 the theory is applied to the liquid-to-solid phase transition and in Section 4 to the vapor-to-liquid phase transition. In Section 5, more complex cases, such as nucleation on soluble particles and aerosols (Sections 5.1 and 5.3), binary nucleation (Section 5.2) and protein folding (Section 5.4) are considered. 2. Nucleation rate During the nucleation stage of a first order phase transition, the new phase appears as a manifold of clusters of various sizes. Denoting by fi the time-dependent number of clusters consisting of i (i = 1, 2, …) molecules, the flux Ii of clusters passing from the population of clusters fi − 1 to population fi is given by [31] þ
−
Ii ¼ W i−1 f i−1 −W i f i
ð1Þ
+ where W− i and Wi are the rates of evaporation (dissociation) and condensation of molecules from and on the surface of clusters belonging to population fi, respectively, both dependent on the cluster size. The change of fi as function of time is provided by the equation
df i þ − þ − ¼ − Iiþ1 −Ii ¼ W i−1 f i−1 − W i þ W i f i þ W iþ1 f iþ1 : dt
ð2Þ
Assuming a smooth dependence of fi and W± i on i, Eq. (2) can be transformed into a partial differential equation which is more convenient for analysis than Eq. (2). That transformation can be performed in various ways which provide slightly different equations. In RNNT, two such transformations are employed. The first one, suggested by Goodrich [32], consists in expanding the continuous functions f(g, t), I(g, t), and W±(g) about the midpoint i + 1/2. As a result, the following differential equation is obtained ∂f ðg; t Þ ∂Iðg; t Þ ¼− ∂t ∂g 1 ∂2 þ ∂ þ − − W ðg Þ þ W ðg Þ f ðg; t Þ− ¼ W ðg Þ−W ðg Þ f ðg; t Þ; 2 ∂g 2 ∂g ð3Þ where the continuous variable g replaces the discrete variable i [8,9,17]. Since the time scales of formation of small unstable clusters are much smaller than the time scale of formation and growth of the critical cluster, the system attains a quasi steady state after a short time.
Please cite this article as: Ruckenstein E, et al, A novel approach to the theory of homogeneous and heterogeneous nucleation, Adv Colloid Interface Sci (2014), http://dx.doi.org/10.1016/j.cis.2014.10.011
E. Ruckenstein et al. / Advances in Colloid and Interface Science xxx (2014) xxx–xxx
The steady state flux which provides the nucleation rate Js is given by equation 1 d þ − þ − W ðg Þ þ W ðg Þ f s ðg Þ þ W ðg Þ−W ðg Þ f s ðg Þ 2 dg
Js ¼ −
ð4Þ
where fs(g) is the steady state solution of Eq. (3). The functions fs(g) and the flux Js can be obtained by solving Eq. (3) for the boundary conditions fs(g) → 0 for g → ∞ (large clusters) and fs(g) = ρs for g → 1 (smallest clusters, monomers) where ρs is the number density of monomers at infinite distance from the cluster. The first boundary condition takes into account that clusters with very large sizes are absent. The second boundary condition involves the assumption that nucleation does not cause substantial depletion of the monomers and that their number density ρs remains the initial one (in calculations, the limit 1 can be replaced by zero). Finally, the rate of nucleation, Js, is obtained by introducing the solution of Eq. (3) in Eq. (4). One thus obtains Js ¼ Z
∞ 0
þ
W ð1Þρs
1 2
exp½−2wðg Þdg
Z wðg Þ ¼
þ
−
W ðg Þ−W ðg Þ dg : þ − 0 W ðg Þ þ W ðg Þ g
Z eq þ J s ¼ n1 W ð1Þ=
ν
ð7Þ
exp
1
W þ ðν Þ dν −hðνÞ−ln þ W ð1Þ
þ
W ð j−1Þ : − j¼2 W ð jÞ
exp½hðν Þ ¼ ∏
ð11Þ
ð12Þ
The function −hðν Þ−ln
W þ ðν Þ ≡ H ðνÞ W þ ð1Þ
ð13Þ
in Eq. (11) exhibits a sharp maximum for the critical cluster size ν = ν∗ and the steepest descent method allows to write for the nucleation rate eq
ð6Þ
∞
where function h(ν) is provided by equation
þ
J s ≃n1 W ð1Þ −
It should be noted that in Eq. (6), the rates of dissociation, W−(g), and condensation, W+(g), while functions of the discrete number of molecules in the cluster, are considered as continuous functions of g, or, equivalently, of the cluster radius R. The function w(g) exhibits a sharp minimum at the critical cluster size g = g∗ and the integral in Eq. (5) can be calculated using the steepest descent method. As a result, the rate of nucleation acquires the form !1=2 1 þ w″ ðg Þ J s ¼ W ð1Þρs exp½2wðg Þ 2 π
Using the stationary solution of the above equation for the boundary eq conditions, f(1)/neq 1 = 1 and f(ν)/n (ν) → 0 as ν → ∞, one obtains the steady-state nucleation rate Js
ð5Þ
where W+(1) is the condensation rate calculated for clusters consisting of one molecule and
3
!1=2 H ″ ðν Þ expð−H ðν ÞÞ: 2π
ð14Þ
Eq. (14) was used for the description of nucleation rate for vapor-toliquid transition. More details about the transformation from the discrete equation Eq. (2) to continuous differential equations are presented in Ref. [11]. For comparison, the steady-state nucleation rate of CNT for vaporto-liquid transition is provided by the equation CNT
Js
eq
¼
þ
n1 W ð1Þ ση2 kB T 4πρη3
!1=2 exp −
! 16πσ 3 3ðkB T Þ3 ρ2 ln2 ðζ Þ
ð15Þ
where η is diameter of vapor molecule, ρ the vapor density, ζ the vapor supersaturation, and σ the liquid–vapor surface tension. 3. Liquid-to-solid phase transition 3.1. Dissociation rate
where w″(g) is the second derivative of w(g) with respect to g. To determine w(g), one should find first the rates W+(g) and W−(g) that is one of the main goals of RNNT. Eqs. (5) and (7) were used in Refs. [8–10] for calculating the nucleation rate in liquid-to-solid transformation. The second type of transformation from the discrete Eq. (2) to a continuous one involves the detailed balance þ eq
−
eq
W i ni ¼ W iþ1 niþ1
ð8Þ
which provides the equilibrium solution of the differential equation obtained from Eq. (2) [33]. In Eq. (8), neq i is the equilibrium number of clusters containing i molecules. Eq. (8) can be applied recursively to obtain eq
eq
W þj−1 − j¼2 W j i
ni ¼ n1 ∏
ð9Þ
where neq 1 is the equilibrium number of clusters consisting of single molecules. Expressing the discrete cluster sizes by the continuous variable ν and combining Eq. (2) with Eq. (8), the evolution of the cluster distribution f(ν, t) can be described by (see Refs. [11,13]) ∂f ðν; t Þ ∂ ∂ f ðν; t Þ ∂ þ eq ¼ − Iðν; t Þ: ¼ W ðν Þn ðν Þ eq ∂t ∂ν ∂ν n ðν Þ ∂ν
ð10Þ
Let us consider a spherical cluster of a solid phase surrounded by a liquid. The molecules of the cluster are held together by attractive forces. Because of collisions with the molecules of the surrounding medium, a surface monomer exhibits Brownian motion in the attractive potential well and dissociates from the cluster as soon as it surpasses the range of the interaction potential. The rate of dissociation of surface monomers is provided by the kinetic law of a first order reaction W
−
¼
Ns bτN
ð16Þ
where Ns is the number of surface monomers and b τ N is the mean first passage time, characterizing the average time required for a surface monomer to cross the outer boundary of the potential well. The Brownian motion of a molecule in the potential well is described by a Fokker–Plank equation for the joint probability density function f(r, v, t) in the phase space of position r and velocity v as [7] ∂f ∇Φ k T ∇v f ¼ γ f ∇v v f þ B ∇v f ; þ v ∇r f − m m ∂t
ð17Þ
where Φ(r) is the interaction potential between a molecule of liquid and those of the cluster, m is the mass of a surface monomer, γf is the friction coefficient with the surrounding medium. Because in liquids the relaxation time to achieve equilibrium velocity distribution (which is of the order of γ−1 f ) is much shorter than the dissociation time, the
Please cite this article as: Ruckenstein E, et al, A novel approach to the theory of homogeneous and heterogeneous nucleation, Adv Colloid Interface Sci (2014), http://dx.doi.org/10.1016/j.cis.2014.10.011
4
E. Ruckenstein et al. / Advances in Colloid and Interface Science xxx (2014) xxx–xxx
Fokker–Plank equation reduces to the Smoluchowski equation, which describes diffusion in an external field [34,35]. In the case of spherical symmetry, this equation has the form ∂nðr; t Þ −2 ∂ 2 −Φðr Þ ∂ Φðr Þ ¼ Dr r e e nðr; t Þ ∂t ∂r ∂r
ð18Þ
where n(r, t) is the number density of diffusing surface monomers in their potential wells at position r and time t, D the diffusion coefficient, r the radial distance from the center of symmetry, and Φðr Þ ¼ Φðr Þ=kB T. To determine b τ N, the first passage time τ(r0) for a molecule that is initially located at the distance r0 from the center of the cluster must be first calculated and then averaged by integrating over r0 for the initial distribution n(r0, 0) of molecules. For calculations, it is more convenient to deal with the backward evolution of the conditional probability density w(r, t|r0, 0) which provides the probability for the monomer to be located at the final (fixed) position r at time t given that its initial position is r0. The corresponding backward Fokker–Planck (backward Smoluchowski) equation is given by ∂ −2 Φðr Þ ∂ 2 −Φðr 0 Þ ∂ r0 e wðr; tjr 0 ; 0Þ : wðr; tjr 0 ; 0Þ ¼ Dr0 e 0 ∂t ∂r 0 ∂r 0
ð19Þ
Z
Rþη R−η
wðr; tjr 0 ; 0Þdr :
τðR þ ηÞ ¼ 0
whereas the boundary condition Eq. (23) can be rewritten as a reflecting boundary condition
dτ
¼0 : dr ðR−ηÞ
The solution of Eq. (25) for the above boundary conditions yields the following expression for τ(r0) τðr 0 Þ ¼
The above expression involves the assumption that the interaction potential is symmetric around the potential well located at R. Integrating Eq. (19) with respect to r one obtains, ð21Þ
If the initial position of the monomer is (R + η), it will dissociate instantaneously and consequently [7,35], ð22Þ
for all t. The radial distance of the surface monomer cannot be less than (R − η) because of the hard sphere repulsion with the neighboring monomers. Consequently, the flux of the monomer at (R − η) is zero [7,35], i.e. ∂ Q ðr; t Þjr¼R−η ¼ 0 : ∂r
ð23Þ
From the definition of Q(r0, t), one can see that the probability for the dissociation time of a monomer to be between 0 and t is 1 − Q(r0, t). The first passage time τ(r0) is therefore given by [7] Z τðr 0 Þ ¼ −
∞ 0
t
∂Q ðr0 ; t Þ dt ¼ ∂t
Z
∞ 0
Q ðr 0 ; t Þdt:
ð24Þ
Integrating Eq. (21) with respect to t and taking into account that Q(r0, ∞) = 0, one obtains −2 Φðr Þ ∂ 2 −Φðr 0 Þ ∂ r0 e τðr 0 Þ ¼ 1 : −Dr 0 e 0 ∂r 0 ∂r 0
1 D
Z
Rþη
x
−2 ΦðxÞ
ð25Þ
Z
x
2 −ΦðxÞ
e
y e ðR−ηÞ
0
dydx:
ð28Þ
Therefore, the average first passage time b τ N for all surface monomers at any initial location can be obtained by averaging Eq. (28) with respect to the equilibrium distribution of surface monomers within the potential well given by, =Z
ð29Þ
where Z is the partition function Z Z¼
Rþη
2 −ΦðxÞ
x e R−η
dx :
ð30Þ
One thus obtains Z bτN ¼
Q ðR þ η; t Þ ¼ 0
ð27Þ
2 −ΦðxÞ
ð20Þ
∂ −2 Φðr Þ ∂ 2 −Φðr0 Þ ∂ r0 e Q ðr 0 ; t Þ : Q ðr 0 ; t Þ ¼ Dr 0 e 0 ∂t ∂r 0 ∂r 0
ð26Þ
peq ðxÞ ¼ x e
A surface monomer is associated with the cluster as long as its radial position is less than (R + η), η being the range of the interaction potential. The probability Q(r0, t) for the dissociation time of the surface monomer to be greater than t provided that its initial position is r0, is given by Q ðr0 ; t Þ ¼
The boundary condition Eq. (22) can be rewritten as an absorbing boundary condition
Rþη R−η
τðxÞpeq ðxÞdx :
ð31Þ
3.2. Condensation rate The condensation rate W+ is obtained from the stationary solution of the diffusion equation in an external field and has the form þ
W ¼ γ4πDRρs
ð32Þ ∞
−2 Φðr Þ
where γ−1 ¼ R∫Rþη r e dr. Here and below, the variable g (number of molecules in the cluster) in W±(g) is omitted. The relation between g and R is provided by the expression g ¼ 4π3 ρc R3 where ρc is the number density of molecules in the cluster. The rate of condensation is affected (through γ) by the potential field Φ(r). Note that in the limit R → ∞, the factor γ tends to unity and Eq. (32) leads to an expression for the condensation rate in the absence of an external field [9]. The size Rc of the critical nucleus can be calculated by using the equality between the dissociation and condensation rates (W+ = W−) provided by Eqs. (16) and (32). 3.3. Results Applications of RNNT to liquid-to-solid nucleation were carried out for amorphous spherical clusters (Refs. [8,9,28]), and fcc and icosahedral clusters (Ref. [10]). In general, the cluster (with number density ρc) is surrounded by molecules of solvent (number density ρw) and molecules of solute (number density ρs). The interaction potential between any two molecules was selected as a combination of a hard core repulsion and dispersion attraction ϕðr 12 Þ
¼
8
