Article pubs.acs.org/JPCB
Mobile Monomers and Dimers in Precipitation Kinetics: a Microscopic Approach Gersh O. Berim, Lana I. Brim, and Eli Ruckenstein* Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260, United States ABSTRACT: A microscopic theory of precipitation kinetics in solution developed previously by Ruckenstein and co-workers [Dadyburjor, D. B.; Ruckenstein, E. J. Cryst. Growth 1977, 40, 279−290; Bhakta, A.; Ruckenstein, E. J. Chem. Phys. 1995, 103, 7120−7135] is generalized. The processes (not considered in the original approach) of monomer−monomer agglomeration, leading to the creation of dimers, as well as absorption (emission) of dimers by solute particles due to dimer mobility are included in the theory. The theory is applied to a model system in which particles grow up to a certain largest size and then precipitate from solution. The most important change in the system kinetics due to those two processes (monomer agglomeration to form dimers and dimer absorption and emission) is tremendous slowing of the asymptotic time behavior of the concentration of particles of largest size. This can be used to obtain experimental evidence for agglomeration of monomers and dimer mobility in the kinetics of real systems. The effect of trimer absorption (emission) is estimated, and it is shown that it is negligible in many situations.
1. INTRODUCTION In refs 1, 2, a microscopic theory describing the growth of nuclei of a new phase during phase transformations was developed by Ruckenstein et al. The theory, which will be referred to below as Ruckenstein−Dadyburjor−Bhakta theory (RDBT), was applied to describe Ostwald ripening, a final stage of phase separation in various supersaturated solutions (e.g., solid−solid, liquid−liquid, solid−liquid, liquid-vapor, etc.), during which particles (clusters) of a solute are formed and grow unrestricted as long as the solution remains supersaturated. However, situations are possible in which the growth of particles is restricted by a finite size, jmax. (The size of a particle is characterized by the number j of monomers it contains.) As soon as the size of a particle becomes jmax, it precipitates from the system (it no longer absorbs or emits monomers). A few examples are denitrification of nitric acid dihydrate in the polar stratosphere3 and crystallization of some organic molecules (e.g., sugars) from supersaturated solutions.4 In the present article, the main equations of RDBT are modified to describe such situations. Along with this, the theory is extended by modifying two important assumptions that were used in refs 1, 2. The first assumption was that particle growth (decay) occurs only by the absorption (emission) of monomers. The possible contributions of the absorption (emission) of larger particles (dimers, trimers, etc.) were neglected. This approximation is valid if the mobility of larger particles due to their diffusion motion is negligible compared to that of monomers or the rate of absorption (emission) of larger particles is much smaller than that of monomers. The second assumption was that monomer−monomer agglomeration, © 2017 American Chemical Society
which leads to the appearance of a large number of dimers in the system, can be neglected. As a possible reason for this, the authors considered the weakness of the monomer−monomer interaction compared with the monomer−particle one. However, there is theoretical5 and experimental6,7 evidence that in some situations both of those assumptions are not applicable. In particular, the diffusion coefficient, D, of spherical particles of radius r in a liquid of viscosity η, provided by the Stokes−Einstein equation
D=
kBT 6πηr
(1)
(kB is the Boltzmann constant and T is the absolute temperature), decreases slowly with increasing r. For example, the ratio of diffusion coefficients for dimers and monomers is equal, 1/ 3 2 ≃ 0.79, and cannot be considered small. In addition, the processes of monomer−monomer agglomeration and absorption (emission) of dimers by larger particles were detected experimentally in ref 8. For these reasons, we have taken into account mobility of dimers, leading to their absorption (emission) by larger particles, and monomer−monomer agglomeration. Particles with j > 2 are considered motionless (no diffusion), and their absorption (emission) by other particles is neglected. However, those particles can absorb or emit monomers and dimers. Received: October 19, 2016 Revised: December 12, 2016 Published: January 9, 2017 854
DOI: 10.1021/acs.jpcb.6b10573 J. Phys. Chem. B 2017, 121, 854−862
Article
The Journal of Physical Chemistry B
where D is the diffusion coefficient of the monomers. When the total number of molecules in the system is conserved, system eq 2 has to be complemented by the equation
In Section 2, the main equations and procedures of RDBT are reviewed. They are modified in Section 3 by accounting for monomer−monomer agglomeration and dimer mobility. In Section 4, the results for time dependencies of concentrations of free monomers and particles with 2 ≤ j ≤ jmax as well as of the critical radius are obtained on the basis of the original RDBT (Section 4.1) and an extended RDBT (Sections 4.2−4.4). Each section contains a brief discussion of the specific results. General trends in the system’s kinetics due to monomer agglomeration and dimer activity are discussed in Section 5.
jmax
N0 = c1 +
(j ≥ 2)
where N0 is the constant total number of solute molecules per unit volume and jmax is the size of the largest possible particle in the system. The critical radius of the particle, rc, is defined as the radius of the particle for which the rate of emission of monomers is equal to the rate of their absorption, that is, L+j = L−j . The latter equation is equivalent to the equation csj = c0j. Using the relation between the size, j, of the particle and its radius, rj
(2)
j=
where cj is the concentration (molecules per unit volume) of the spherical particles comprising j monomers and L+j and L−j are the rates of growth and decay of those particles due to absorption and emission of a monomer, respectively. It was assumed that there was no agglomeration of monomers (L+1 = 0) and that only monomers can be attached to or detached from the particle. Rate L+j was considered to be proportional to the surface area of a particle and to the actual monomer concentration, c0j, at the interface of a particle of size j. Rate L−j was assumed to be proportional to the surface area of the particle and to the concentration, csj, of monomers in equilibrium with a particle of size j. Thus Lj+ = 4πr j2kc 0j , L−j = 4πr j2kcsj ,
(j > 2)
rc =
3vm
(8)
δ ln(c1/cs∞)
(9)
The critical size, jc, can be obtained by replacing the radius, rj, in eq 8 by rc from eq 9. Note that the ratio c1* = c1/cs∞ is the supersaturation ratio. If the system is in a supersaturated state, the actual concentration of the monomers, c1, is greater than cs∞ (c*1 > 1).
(3)
3. EXTENDED RDBT 3.1. Main Equations. To take into account the agglomeration of monomers and growth (decay) of particles due to absorption (emission) of dimers in the framework of the considered model, eq 2 is modified as follows d − + c 2 = L1+c1 + L3−c3 − L 2+c 2 − L 2−c 2 + Ld,4 c4 − Ld,2 c2 dt d cj = Lj+− 1cj − 1 + Lj−+ 1cj + 1 − Lj+cj − Lj−cj + Ld,+j − 2cj − 2 dt + Ld,−j + 2cj + 2 − Ld,+jcj − Ld,−jcj , (3 ≤ j ≤ jmax − 3) d c j − 2 = L j+ − 3c j − 3 + L −j − 1c j − 1 − L j+ − 2c j − 2 max max max max max max dt max − L −j − 2c j − 2 + Ld,+j − 4c j − 4 − Ld,+j − 2c j − 2
(4)
where cs∞ is the concentration of the solute in equilibrium with a large particle (rj → ∞) and δ is given by the equation 2γvm δ≡ kBT
4πr j3
along with eqs 4 and 6, one can finally obtain for critical radius rc the expression
where rj is radius of a spherical particle containing j monomers and k is the rate constant, which is taken to be the same for L+j and L−j and is assumed to be independent of j. As reported previously, the particles are considered spherical with smooth surfaces. In reality, those surfaces can have more complicated structures. For example, in ref 9, it was shown that fractal-like surfaces are possible. To take this into account, a special approach must be developed. Concentration csj is provided by the Gibbs−Thomson formula ⎛δ⎞ csj = cs∞ exp⎜⎜ ⎟⎟ ⎝ rj ⎠
(7)
2
2. ORIGINAL RDBT In its general form, the main kinetic equation of RDBT can be written as2 d cj = Lj+− 1cj − 1 + Lj−+ 1cj + 1 − Lj+cj − Lj−cj , dt
∑ jcj
max
− Ld,−j
max
max
− 2c j
max
max
max
max
max
−2
d c j − 1 = L j+ − 2c j − 2 − L j+ − 1c j − 1 − L j− − 1c j − 1 max max max max max max dt max + Ld,+j − 3c j − 3 − Ld,−j − 1c j − 1
(5)
In eq 5, γ is the interfacial tension between the solid and liquid phases and vm is the mean volume of monomers of the solute. Concentration c0j was calculated in ref 2 from the consideration of a quasisteady state material balance at the particle interface, assuming that the monomer motion is due to diffusion. The result is provided by the formula2 c1 − csj c 0j = csj + krj 1+ D (6)
max
max
max
max
d c j = L j+ − 1c j − 1 + Ld,+j − 2c j − 2 max max max max dt max (10)
L+d,j
L−d,j
where and are the rates of growth and decay of a particle of size j due to the gain and loss of a dimer, respectively. Note the specific forms of the three last differential equations (for d d d c , c , and dt c j ) in eq 10, which are written for dt j − 2 dt j − 1 max
855
max
max
DOI: 10.1021/acs.jpcb.6b10573 J. Phys. Chem. B 2017, 121, 854−862
Article
The Journal of Physical Chemistry B the case in which the largest particles containing jmax monomers do not loose or gain monomers or dimers. Rates L+d,j and L−d,j are selected similarly to L+j and L−j in the form Ld,+j = 4πr j2kdd 0j , Ld,−j = 4πr j2kddsj ,
(j ≥ 2)
d * c 2 = k1*r1*2c 0*1c1* + r3*2cs*3c3* − r2*2c 0*2c 2* − r2*2cs*2c 2* dt * + kd*(r4*2ds*4 c4* − r2*2d 0*2c 2*) d * c j = r *j −21c 0*j−1c*j − 1 + r *j +21cs*j+1c*j + 1 − r *j 2c 0jc*j − r *j 2cs*j c*j dt * + kd*(r *j −22d 0*j−2c*j − 2 + r *j +22dsj+2c*j + 2 − r *j 2d 0jc*j − r *j 2ds*j c*j )
(11)
where kd is the rate constant for the dimers and d0j and dsj are
,
(similar to c0j and csj) the actual dimer concentration at the particle interface and the concentration of the dimers in equilibrium with a particle of size j, respectively. For d0j and dsj,
d * c j − 2 = r *j 2 − 3c 0*j −3c *j − 3 + r *j 2 − 1cs*j −1c *j − 1 max max max max max max dt * max − r *j 2 − 2c 0*j −2c *j − 2 − r *j 2 − 2cs*j −2c *j − 2 max
formulas similar to those for d0j and dsj are considered d 0j = dsj +
max
max
kdrj
max
max
1 + krj/D
+ 2kd
1 + kdrj/Dd
max
−2
− 2)
max
max
max
max
max
c* −1 j
− 1) max
∑ jc*j (18)
j=2
where N*0 ≡ N0/cs∞ represents the total number of monomers per unit volume in the system. 3.2. Initial Conditions and Parameters of the Model. The initial distribution of concentrations, c*j , needed for solving eq 17 is the result of the nucleation stage of phase transformation. It is selected in the same form as in ref 2 ⎡ (j − j )2 ⎤ 0 ⎥ c*j |t *= 0 = c *j exp⎢ − , 0 ⎢⎣ 2s 2 ⎥⎦
=0 (15)
(j ≥ 2) (19)
where j0 is the number of monomers in particles that are most numerous (with concentration c*j0 ) in the initial state and s is the standard deviation. Together with the conservation equation (eq 7) and initial condition (eq 19), eq 17 provides a complete description of the system kinetics. To avoid lengthy calculations, the size of the largest particles in the system was selected to be comparatively small (jmax = 350). The system of eq 17 was used to calculate the time behavior of the concentrations of free monomers (c*1 (t*)); dimers (c2*(t*)); particles with size j = 200 (c200 * (t*)), representing concentrations of particles with 2 < j < jmax; and the largest particles (cj*max(t*)). In addition, the time dependence of the critical radius, rc*(t*), was also calculated for several cases. In the calculations, the initial dimensionless concentration of the free monomers, c*1 (0), is selected to be the same for all considered cases (c1*(0) = 1.10). In the absence of dimer contribution (kd* = 0), the critical radius at t* = 0 is rc,0 * = 10.4,
where csj and dsj are provided by eqs 4 and 13. Because concentrations c1 and c2 in eqs 9 and 15 depend on time, the critical radius (critical size) is also time-dependent. To take into consideration agglomeration of monomers, which leads to an increase in the number of dimers, the term L+1 c1 was added to the equation for c2 in system eq 10. The rate of agglomeration of monomers, L+1 , was selected in the form L1+ = 4πr12k1c 01
c *j
max − 2
jmax
N0* = c1* +
where vm,d is the mean radius of the dimers. For simplicity, it was assumed that vm,d = 2vm. The possibility of growth (decay) of the particles due to dimer mobility changes the equation for critical radius (critical size) of the particle. The rates of growth and decay of a particle of radius rj containing j monomers are equal now: L+j + 2L+d,j and L−j + 2L−d,j, respectively. The equality of those rates provides the following equation for rc k
max
Here, all concentrations are in units of cs∞, r*j ≡ rj/δ, k*1 ≡ k1/k, kd* ≡ kd/k, and t* ≡ 4πkδ2cs∞t. Equation 7 takes the following dimensionless form
(14)
c 2 − d sj
−4
(17)
2γvm,d kBT
max
max
max
max
− r *j 2 − 2d 0*j
d * c j = r *j 2 − 1c 0*j −1c *j − 1 + kd*r *j 2 − 2d 0*j −2c *j − 2 max max max max max max dt * max
(13)
where d0∞ is the concentration of the dimers in equilibrium with a large particle (rj → ∞). Parameter δd in eq 13 is provided by the equation
c1 − csj
max
− r *j 2 − 1ds*j
⎛δ ⎞ dsj = ds∞ exp⎜⎜ d ⎟⎟ ⎝ rj ⎠
c *j
max − 2
max
c* −4 j
d * c j − 1 = r *j 2 − 2c 0*j −2c *j − 2 − r *j 2 − 1c 0*j −1c *j − 1 max max max max max max dt * max − r *j 2 − 1cs*j −1c *j − 1 + kd*(r *j 2 − 3d 0*j −3c *j − 3
(12)
where Dd is the diffusion coefficient of the dimers and
δd ≡
max
− r *j 2 − 2ds*j
Dd
max
max
+ kd*(r *j 2 − 4d 0*j
c 2 − d sj 1+
j = 3, 4, ..., jmax − 3
(16)
where rate constant k1 may be different from k. For convenience of the numerical calculations, the final system of kinetic equations was rewritten in the dimensionless form, as follows 856
DOI: 10.1021/acs.jpcb.6b10573 J. Phys. Chem. B 2017, 121, 854−862
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The Journal of Physical Chemistry B which corresponds to the initial critical size, jc,0 = 106. The initial concentrations of particles with j ≥ 2 (eq 19) were selected to provide the total number, N0*, of monomers (free monomers plus those in particles) per unit volume greater than c*1 (0) (N*0 = 32.1). The value s = 5 was used for standard deviation. The other necessary parameters were selected as follows
supersaturated state, that is, the solute is never depleted. Second, at t* → ∞, concentration c*jmax(t*) of the largest particles, with j = jmax in the case of j0 = 35, is not zero and coincides with that calculated for the cases in which j0 = 105 and 175 (c*jmax(∞) ≃ 0.089). Such a result is not obvious because at j0 = 35 the initial state does not contain particles with size j greater than the critical size, jc,0, and according to the conventional point of view10 all existing particles should disappear by successive emission of monomers. Hence, concentration cj*max(∞) is expected to be zero. We will discuss these two unusual features below along with consideration of the time behavior of the critical radius, which is presented in Figure 2 for all three considered cases. From the latter figure,
δd = 2δ , Dd /D = 0.1, ds∞/cs∞ = 0.2, kδ /D = 0.01, vm /δ 3 = 44.7
(20)
The time dependence of the concentrations was examined in the time interval 0 ≤ t* ≤ 106. The asymptotic behavior at t* → ∞ is estimated using t* → 106. Note that the ratio Dd/D was intentionally selected to be much smaller than that predicted by the Stokes−Einstein equation (eq 1) (Dd/D ≃ 0.79) to show that even a small diffusion coefficient for dimers can lead to significant changes in the system kinetics.
4. RESULTS To estimate (at least qualitatively) the contributions of monomer agglomeration and dimer mobility, the quantities of interest will first be calculated, in Section 4.1, on the basis of the original RDBT (k*d = k*1 = 0). The same quantities are further calculated using the modified RDBT (Sections 4.2−4.4) and compared to the former ones. In all cases, a brief discussion of specific details of the time behavior of the considered quantities is provided. The general trends in the system kinetics are discussed in Section 5. 4.1. Original RDBT. To examine the role of the initial size distribution in system kinetics, parameter j0 was selected to be smaller than the initial critical size (j0 = 35), about the same size as it (j0 = 105), or larger (j0 = 175) than the initial critical size of the particles. In Figure 1, the time dependencies of the concentrations of monomers and particles of various sizes at different initial
Figure 2. Time dependencies of critical radius r*c (t*) at various values of j0: j0 = 175 (solid line), 105 (dashed line), and 35 (dotted line).
one can see that the critical radius steadily increases for j0 = 105 with increasing t*. For j0 = 35, it decreases rapidly at small times; passes through the minimum, r*c,min ≃ 7.2 (jc,min ≃ 35); and then increases. For j0 = 175, the critical radius first quickly increases up to rc,max * ≃ 12.4 (jc,max ≃ 179) and then slowly decreases. In all cases, the critical radius approaches the same asymptotic value, r*c (∞) ≃ 11.7, that corresponds to critical size jc,∞ = 149. To explain such a difference in the time behavior of rc*(t*) at different j0, let us note that the critical radius is determined by the concentration, c*1 (t*), of free monomers in the system (see eq 9). It decreases (increases) with increasing (decreasing) c1*(t*). At j0 = 175, almost all particles in the initial state have a size greater than the critical size (jc,0 ≃ 106) and they tend to grow at t* > 0 due to the absorption of free monomers. Therefore, the concentration of free monomers in the system decreases (see Figure 1a) and, as a consequence, the critical radius increases and passes through a maximum. The decrease in r*c (t*) after reaching a maximum can be assigned to the emission of monomers (leading to an increase in c1*(t*)) from particles with sizes smaller than rc*(t*). At j0 = 35, almost all particles have a size smaller than the critical size and they start to decay by releasing free monomers. Because of this, the concentration of free monomers increases (Figure 1a) and the critical size decreases up to jc,min ≃ 35 and becomes comparable to j0. This allows the growth of particles with size j ≥ jc,min, which explains the above-mentioned nonzero value of c*jmax(∞) for j0 = 35. As one can see from Figure 1a, for j0 = 35, after passing a maximum, the number of free monomers decreases and approaches its asymptotic value, c*1 (∞) = 1.11. In turn, the critical radius increases after passing a minimum and approaches the value rc*(∞) = 11.7.
Figure 1. (a−d) Time dependencies of c*1 (t*), c*2 (t*), c*200(t*), and c*jmax(t*), respectively, for the case k*d = k*1 = 0 (original RDBT) at various values of j0: j0 = 175 (solid line), 105 (dashed line), and 35 (dotted line).
distributions are presented. Several unexpected features of those time dependencies should be noted. First, for all considered cases, the dimensionless concentrations of the monomers remain greater than 1 (c1*(t*) > 1) at any time, including t* → ∞. This means that the system is always in the 857
DOI: 10.1021/acs.jpcb.6b10573 J. Phys. Chem. B 2017, 121, 854−862
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The Journal of Physical Chemistry B
At k1* ≠ 0, the time dependence of free-monomer concentration, c*1 (t*), is more complicated than that in the case of k1* = 0 (see Figure 3a−c for the former case and Figure 1a (dashed line) for the latter one). At a small t*, c1*(t) quickly decreases, passes through a minimum with the depth dependent on k*1 , then increases, passes through a broad maximum (Figure 3b), and then approaches (see Figure 3c) the final value, which is the same for all nonzero k1* (c1*(∞) ≃ 1.04). Because c*1 (∞) ≥ 1, the system, at t* → ∞, remains in the supersaturated state, as in the case of k*1 = k*d = 0. The concentrations of particles with j = jmax approach different asymptotic values at different k1*, which decrease with increasing k1*. This happens because of an increase in concentration c2*(t*) with increasing k1*. The larger the c2*(t*) value, the smaller the c1*(t*) value and, as a consequence, the smaller the concentration of the largest particles, cj*max(t*), which, in case A, grows only because of the absorption of monomers by smaller particles. Numerical analysis shows that at t* → ∞ the results for cj*max(t*) at k1* ≠ 0 can be very well approximated by the formula
In the intermediate case (j0 = 105), at t* = 0, approximately one half of the particles with j ≥ 2 have radii smaller than r*c,0 and the other half have radii larger than r*c,0. In this case, the time behavior of r*c (t*) and c*1 (t) is shown by dashed lines in Figures 2 and 1a, respectively. In conclusion to this section, let us note that the time behavior of c*jmax(t*) at a large t* can be approximated by the formula c *j (t *) ≃ c *j (∞) − a *j exp( −t */τ *) max
max
max
(21)
with cj*max(∞) = 0.089, a*jmax = 0.19, and τ* = 32.8. A similar exponential asymptotic behavior at t* → ∞ includes all other calculated quantities (c*1 (t*), c*2 (t*), c*200(t*)). 4.2. Modified RDBT: Case A (kd* = 0, k1* ≠ 0). To examine the role of monomer agglomeration, let us consider the case in which kd* = 0 (no dimer mobility) and k1* varies. This case will be referred to as case A below. The initial particle size distribution is selected for j0 = 105. Note that the results obtained for initial distributions with other values of j0 and s differ from those at j0 = 105 and s = 5 only for small times and practically coincide with them in the most interesting case of t* → ∞.
c *j (t *) ≃ c *j (∞) − bd*/ log10 t *, max
max
(k1* > 0)
(22)
which indicates a very slow nonexponential approach of c*jmax(t*) to its asymptotic value c*jmax(∞). The values of c*jmax(∞) and b*d at various k1* are presented in Table 1. Table 1. Values of c*jmax(∞) and bd* in Equation 22 for Various k1* at kd* = 0 k*1 cj*max(∞)
0.1 0.088
0.2 0.087
0.4 0.085
0.6 0.082
0.8 0.080
1.0 0.078
b*d
0.0081
0.016
0.030
0.044
0.059
0.073
The time dependence of the critical radius at various k1* values is provided in Figure 4. At small times, rc*(t*) increases with increasing k*1 , the rate of increase being larger for larger k*1 (Figure 4a). For k*1 = 0.1, 0.2, and 0.4, the critical radius passes through a maximum (the three lowest curves in Figure 4b) and then slowly decreases (Figure 4c). In the cases of k1* = 0.6, 0.8, and 1.0, r*c (t*) diverges at small times (Figure 4a). Later, it becomes finite again (Figure 4b) and approaches the same
Figure 3. Time dependencies of c*1 (t*) (a−c), c*2 (t*) (d), c*200(t*) (e), and c*jmax(t*) (f) at k*d = 0 and k*1 = 0, 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0. In (d), concentration c*2 (t*) for the case k*1 = 0 is multiplied by factor 75. In the initial distribution, j0 = 105. In all panels, k1* increases in the direction of the arrow.
In Figure 3, results are presented for the time dependencies * (t*), and c* of concentrations c1*(t*), c2*(t*), c200 jmax(t*) at kd* = 0 and k1* = 0, 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0. Comparing Figures 3 and 1, one can see that the most significant difference between the k*1 ≠ 0 and k*1 = 0 cases is that in the former case there is a large concentration of dimers at t* → ∞ (Figure 3d), whereas in the latter case, this concentration approaches 0 (Figure 1b). The concentration of the dimers increases with increasing k*1 and varies at large times from c2*|k1*=0.1 = 0.048 to c2*|k1*=1 = 0.48.
Figure 4. Time dependencies of rc*(t*) at kd* = 0 and k1* = 0, 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0 in the intervals 0 ≤ t* ≤ 0.05 (a), 0 ≤ t* ≤ 1 (b), 1 ≤ t* ≤ 20 (c), and 20 ≤ t* ≤ 106 (d). In the latter case, the curves for k*1 > 0 are indistinguishable. In the initial distribution, j0 = 105. In all panels, k1* increases in the direction of the arrow. 858
DOI: 10.1021/acs.jpcb.6b10573 J. Phys. Chem. B 2017, 121, 854−862
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The Journal of Physical Chemistry B asymptotic value as that in the case of k*1 ≤ 0.4 at t* → ∞. All of these specific features can be easily explained considering the time dependence of concentration c1*(t*) of free monomers, which, according to eq 9, defines the critical size. This dependence is presented in Figure 3a−c. One can see that for k*1 ≥ 0.4 c*1 (t*) is less than 1 (c*1 (t*) < 1) in some time interval, for example, the system is undersaturated during that time and the critical radius is not defined. For k1* ≤ 0.4, c1*(t) is larger than 1 at all times and has a minimum (see Figure 3a−c). The presence of that minimum provides the maximum in the time behavior of rc*(t) (Figure 4b). 4.3. Modified RDBT: Case B (kd* ≠ 0, k1* = 0). To examine the role of dimers, the rate of agglomeration of monomers was selected to be 0 (k*1 = 0) and the quantities presented in Figures 3 and 4 were calculated for various kd* (see Figures 5
Figure 5. (a−d) Time dependencies of concentrations c1*, c2*, c200 * , and c*jmax at k1* = 0 and kd* = 0, 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0. The initial distribution is that with j0 = 105. In all panels, kd* increases in the direction of the arrow.
Figure 6. Time dependencies of the critical radius at k*1 = 0 and k*d = 0, 0.02, 0.04, 0.06, 0.08, 0.10, and 0.12 in the intervals 0 ≤ t* ≤ 0.15 (a), 0.15 ≤ t* ≤ 80 (b), and 80 ≤ t* ≤ 8000 (c). The initial distribution is that with j0 = 105 for all considered cases. Arrows point in the direction of increase of k*d .
and 6). This case will be referred to as case B below. Comparing Figures 3 and 5, one can see that in both cases * (t*) is similar, whereas other calculated concentration c200 quantities show significant differences. (i) The concentration of dimers, c2*(t*), is much larger in case A (Figure 3d) than that in case B (Figure 5b). This occurs because in case A the dimers, which are created by monomer agglomeration, cannot be absorbed by particles and their concentration remains high. In case B, dimers are not created by agglomeration of monomers but can appear only as a result of emission of monomers from trimers or emission of dimers from larger particles. Because the concentrations of those particles are small, this mechanism also does not generate a large number of dimers compared with agglomeration of monomers. In addition, in case B, dimers can be absorbed; hence, their concentration tends to decrease compared to that in case A. (ii) The time behaviors of the concentration of free monomers, c*1 (t*), and that of particles with j = jmax are different in cases A and B. In case A, c1*(t*) first decreases with increasing time, passes through a minimum, then increases and passes through a maximum, and finally decreases, whereas in case B, it first increases with increasing time, passes through a maximum, and then approaches an asymptotic value, the
latter being larger than that in case A (compare Figures 3a−c with Figure 5a). In case A, the time dependence of c*1 (t*) at a small time is governed by three processes. The first is the increase in c*1 (t*) due to emission of monomers from particles with 2 ≤ j < jc. The second and third processes are a decrease in c1*(t*) due to absorption of monomers by particles with j > 1 and due to agglomeration of monomers, respectively. At all considered values of k*1 ≠ 0, the last two processes are more effective, and they lead to the decrease in c*1 (t*). In case B, agglomeration of monomers is not present. As a consequence, c1*(t*) increases at small times. At a large time (t* → ∞), the variation of cj*max(t*) with a change in k*1 (case A) is much larger than that with a change in k*d (case B; compare Figures 3f and 5d). For instance, in case A, at t* = 106, cj*max(t*) decreases from cj*max(t*) = 8.85 × 10−2 at k1* = 0 to c*jmax(t*) = 6.61 × 10−2 at k1* = 1.0. When kd* varies from 0 to 1 in case B, concentration c*jmax(t*) changes from 8.85 × 10−2 at k*d = 0 to 8.46 × 10−2 at k*d = 1. As in case A, asymptotic behavior of c*jmax(t*) at t* → ∞ is described by eq 22, with parameters c*jmax(∞) and b*d provided in Table 2. 859
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The Journal of Physical Chemistry B Table 2. Values of c*jmax(∞) and bd* in Equation 22 for Various k*d at k*1 = 0 k*d cj*max(∞)
0.1 0.089
0.2 0.088
0.4 0.088
0.6 0.088
0.8 0.087
1.0 0.086
b*d
0.0011
0.0016
0.0031
0.0052
0.0079
0.011
In Figure 6a−c, the time dependence of rc*(t*) is presented for small, intermediate, and large times, respectively, for the case of k*d ≤ 0.12 (k*1 = 0). At a small time, r*c (t*) decreases with increasing time, and for all considered k*d , it becomes close to the same value, rc*(∞) ≃ 10.4 (Figure 6a). At an intermediate time, rc*(t*) increases, the values of rc*(t*) for different k*d being almost the same at any time (Figure 6b). At a larger time, r*c (t*) continues to increase, the rate of increase being larger for larger kd* (Figure 6c). 4.4. Modified RDBT: Case C (k1* ≠ 0 and kd* ≠ 0). To illustrate the case with k*1 ≠ 0 and k*d ≠ 0 (case C), in Figures 7
Figure 8. Time dependencies of concentrations c1* (a−c), c2* (d), c200 * (e), and c*jmax (f) at k*d = 0.4 and k*1 = 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0. The initial distribution is that with j0 = 105. In all panels, k*1 increases in the direction of the arrow.
Table 3. Values of c*jmax(∞) and b*d in Equation 22 for Various kd* at k1* = 0.4
Figure 7. Time dependencies of concentrations c*1 (a), c*2 (b), c*200 (c), and c*jmax (d) at k*1 = 0.4 and k*d = 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0. The initial distribution is that with j0 = 105. In all panels, k*d increases in the direction of the arrow.
kd* cj*max(∞)
0.1 0.084
0.2 0.084
0.4 0.084
0.6 0.084
0.8 0.084
1.0 0.084
b*d
0.032
0.034
0.040
0.048
0.058
0.069
Table 4. Values of c*jmax(∞) and bd* in Equation 22 for Various k*1 at k*d = 0.4
* (t*), and and 8 the results for concentrations c1*(t*), c2*(t*), c200 cjmax(t*) are presented for the cases with k1* = 0.4 and kd* = 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0 and with k*d = 0.4 and k*1 = 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively. Comparing these results, one can see that they are qualitatively the same and have features similar to those in cases A and B. However there are some quantitative differences, among which one can note that at a fixed time the variation of c2*(t*) and cjmax(t*) with variation of k*d at constant k*1 (k*1 = 0.4) is smaller than that with variation of k*1 at constant k*d (k*d = 0.4). This means that agglomeration of monomers affects the time dependencies of the concentrations of dimers and particles with j = jmax strongly compared with dimer activity. The time dependencies of c*jmax(t) follow eq 22, with the coefficients provided in Tables 3 and 4 for the cases presented in Figures 7 and 8, respectively. One can see that the asymptotic time behavior of c*jmax(t) depends strongly on the rate of monomer agglomeration compared with the rate of dimer absorption (emission).
k*1 cj*max(∞)
0.1 0.087
0.2 0.086
0.4 0.084
0.6 0.083
0.8 0.082
1.0 0.081
b*d
0.013
0.022
0.040
0.058
0.076
0.094
5. DISCUSSION One of the important assumptions in the present article is that we do not consider absorption (emission) of particles with size j ≥ 3 by larger particles as a possible source of particle growth (decay). However, as has already been mentioned in the Introduction, such particles can also contribute to the kinetics of supersaturated solutions. Indeed, according to the Stokes− Einstein equation (eq 1) and eq 8, the diffusion coefficient decreases as 1/ 3 j with increasing size of the particles and cannot be considered small for size j ≤ 1000 compared to that for monomers. Because of this, situations are possible in which processes of absorption (emission) of particles with j > 2 can become important. To estimate the conditions under which one can neglect such processes, we have modified the basic equations to take them into consideration and calculate the time dependence of the concentration of the largest particles, cj*max(t*), at various values of the rate constant, ktr*, of absorption (emission) of trimers. The modification was carried out in the same manner as that for dimers. The diffusion coefficient for trimers was selected as approximately equal to Dd. The results 860
DOI: 10.1021/acs.jpcb.6b10573 J. Phys. Chem. B 2017, 121, 854−862
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The Journal of Physical Chemistry B of the calculations are presented in Figure 9 for k*1 = 0.2, k*d = 0.4, and k*tr = 0, 0.1, 0.2, 0.3, and 0.4. This figure indicates that
system is undersaturated. However, it becomes supersaturated again at t* → ∞. If kd* ≠ 0, the critical radius is determined by eq 15. For the considered cases, the critical radius remains constant at large times (see Figure 6c), which indicates that in this case the system is also in the supersaturated state. (iii) Agglomeration of monomers leads to an increase in the concentration of dimers. It is interesting to note that the absorption (emission) of dimers by larger particles does not lead to a decrease in dimer concentration. The latter even increases with an increase in the rate constant of absorption, kd* (see Figure 5d). This can be explained by the presence of a large number of monomers, which agglomerate and compensate the decrease in the number of dimers due to their absorption. (iv) The concentration, cj*max(∞), of the largest particles at t* → ∞ depends slightly on the rate of agglomeration of monomers and is almost independent of dimer absorption (emission). (Compare Tables 3 and 4.) (v) The main consequence of monomer agglomeration and dimer absorption (emission) is the change in the asymptotic behavior of cj*max from exponential to nonexponential (compare eqs 21 and 22). Observation of such a behavior would be an effective method to check experimentally whether the two examined processes are present in real systems. In conclusion, let us mention that the present theory can be improved in several directions by including the processes of absorption (emission) of particles with j > 2, the dependence of rates k, k1, kd, and so forth on the size and shape of the particles, and by examining Ostwald ripening (case jmax → ∞) beyond the restriction of single-monomer absorption (emission). We will consider these issues in the future.
Figure 9. Time dependencies of concentration c*jmax at ktr* = 0, 0.1, 0.2, 0.3, and 0.4 for k1* = 0.2 and kd*. The initial distribution is that with j0 = 105. k*tr increases in the direction of the arrow.
for all considered cases the trimer contribution is small. This suggests that at least in some range of parameters the absorption of particles with size j ≥ 3 can be neglected. Equation 17 provides the possibility of calculating any characteristic of the behavior of the system (the time behavior of the concentrations of particles, the critical radius, the size distributions of particles, etc.). The obtained results show that agglomeration of monomers and absorption (emission) of dimers lead to several significant changes in the system kinetics compared with the case in which those processes are absent. Some of those changes can be associated with the choice of the initial state of the system (short time behavior of the concentration of the free monomers and critical radius); others (large concentration of dimers, tremendous slowing of the asymptotic behavior of c*jmax at t* → ∞) have a more general origin. Comparing the results obtained in Sections 2 and 4.1−4.4, one can identify several general trends related to the presence of agglomeration of monomers and dimer absorption (emission). (i) The asymptotic values of all considered concentrations and the critical radius at t* → ∞ do not depend on the specific features (mean size of the particles, j0, and standard deviation, s) of the initial size distribution of particles. (ii) In all considered cases with kd* = 0, the asymptotic value of the supersaturation ratio provided by function c*1 (t*) remains greater than 1. This means that the system at t* → ∞ remains in a supersaturated state. This feature is a consequence of the restriction imposed on the size of the particles at the formulation of the problem (the particles of size j = jmax do not participate either in the absorption or in the emission of monomers). Because the concentrations of the large particles approach zero at t* → ∞ (see, e.g., Figure 5e), only small particles and particles with j = jmax still exist in the system. The latter particles cannot absorb monomers or dimers and, as a consequence, the number of free monomers, which determines the supersaturation ratio, does not decrease and remains greater than 1 for the selected parameters. Note that at finite t* concentration c1*(t*) can be less than 1 in some cases (see, e.g., Figure 3a), that is, the
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AUTHOR INFORMATION
Corresponding Author
*E-mail: feaeliru@buffalo.edu. Phone: (716)645-1210. Fax: (716)645-3822. ORCID
Gersh O. Berim: 0000-0002-1147-0038 Notes
The authors declare no competing financial interest.
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