Keywords

1 Introduction and Overview

When a material is brought out of equilibrium by a sudden change of thermodynamic parameters (e.g. temperature T, pressure p, etc.), such that a phase boundary of a phase transition (e.g. the melting/crystallization line T m( p) in the p, T plane) is crossed, the old phase (e.g. the fluid) is metastable, and the new phase (e.g. the crystal) forms by nucleation [1, 2]. Homogeneous nucleation (due to statistical fluctuations) requires to overcome a free energy barrier (Fig. 1), Δ F∗, due to the unfavorable surface free energy contribution. Classical nucleation theory estimates this barrier making two assumptions: (i) The critical nucleus can be described by a spherical droplet, R∗ being its radius. (ii) The interfacial free energy just is 4π R∗2 γ, γ being the interfacial tension of a flat planar interface. However, while these assumptions look rather reasonable for the nucleation of liquid droplets from supersaturated vapor, they make little sense for crystal nucleation: the spherical shape of the nucleus is not consistent with its regular crystal structure, and furthermore γ is not isotropic, but rather depends somewhat on the orientation of the interface relative to the crystal axes. This is demonstrated in Fig. 2 for the model of attractive colloidal particles studied in the present work [3, 4]. This model will be explained in the following section. Here it suffices to know that for weak attraction between the colloidal particles this anisotropy is rather weak (left part of Fig. 2), and hence an almost spherical droplet shape may be expected, while for stronger attraction (right part of Fig. 2) the anisotropy is more noticeable. Then the crystal shape will deviate from a sphere. Each point in Fig. 2a, b took around 4 × 24 h on ∼ 1000 CPUs in parallel. If the interfacial free energy were known for arbitrary interface orientation (and not just for the three choices (111), (110) and (100) displayed in Fig. 2), one could find the equilibrium crystal shape from the Wulff construction [5]. First of all, this procedure is cumbersome, and knowing γ(n) for only three orientations n of the interface, this is only possible approximately. If we could do that, the surface term in Fig. 1 could be written as function of the nucleus volume V as

$$\displaystyle{ F_{\mathrm{surf}}(V ) = V ^{2/3}\int \limits _{ A_{w}}\gamma (\mathbf{n})d\mathbf{s} \equiv A_{w}\bar{\gamma }V ^{2/3}\quad, }$$
(1)

where the surface integral ∫ d s is extended over a crystal having the Wulff shape and unit volume. The corresponding surface area is A w, and \(\bar{\gamma }\) is then an average interfacial tension. Then Δ F in Fig. 1 becomes ( p c is the pressure in the crystal and p l is the pressure in the liquid)

$$\displaystyle{ \varDelta F = -(\,p_{c} - p_{\ell})V + F_{\mathrm{surf}}(V ) = -(\,p_{c} - p_{\ell})V + A_{w}\bar{\gamma }V ^{2/3}\quad, }$$
(2)

and the barrier Δ F∗ occurs for V = V∗ with \(\partial (\varDelta F(V ))/\partial V \mid _{V ^{{\ast}}} = 0\). This yields

$$\displaystyle{ V ^{{\ast}1/3} = \frac{2A_{w}\bar{\gamma }} {3(\,p_{c} - p_{\ell})}\quad,\quad \varDelta F^{{\ast}} = \frac{1} {3}A_{w}\bar{\gamma }V ^{{\ast}2/3}\quad. }$$
(3)

Now the present study simply exploits the idea [6, 7] to combine both Eqs. (3) as follows and expand the pressures at the coexistence conditions as

$$\displaystyle\begin{array}{rcl} \varDelta F^{{\ast}} = \frac{1} {2}\Big(\,p_{c} - p_{\ell}\Big)V ^{{\ast}}\quad,& &{}\end{array}$$
(4)
$$\displaystyle\begin{array}{rcl} p_{c}& \approx & p_{\text{coex}} + (6/\pi )\eta _{m}(\mu _{c}(\,p_{c}) -\mu _{\text{coex}}), \\ p_{l}& \approx & p_{\text{coex}} + (6/\pi )\eta _{f}(\mu _{l}(\,p_{l}) -\mu _{\text{coex}}),{}\end{array}$$
(5)

with η m(η f) being the packing fractions of the (spherical) colloidal particles where the melting (freezing) sets in. Since in equilibrium the chemical potential for the nucleus coexisting with fluid is homogeneous,

$$\displaystyle{ \mu _{c}(\,p_{c}) =\mu _{l}(\,p_{l}) =\mu \quad, }$$
(6)
Fig. 1
figure 1

Formation free energy contribution of a nucleus Δ F as function of its linear dimension R. In d = 3 dimensions, the volume term is negative and scales like R3, but the interfacial term is positive and scales like R2. Thus a nucleation barrier Δ F∗ for a “critical droplet” with linear dimension R∗ results

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Fig. 2
figure 2

Finite size scaling for the reduced interfacial tension of the soft effective Asakura-Oosawa (softEffAO) model at two reduced interaction strengths, η p r = 0. 1 (left part) and η p r = 0. 2 (right part), plotted vs inverse interfacial area, using L x × L y × L z geometry and periodic boundary conditions. Three orientations of the interface are shown, (111) [i.e. a closed packed interface in the face-centered cubic crystal lattice], (110) and (100) (Part (a ) is taken from Ref. [3], Part (b ) from Ref. [4])

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Eqs. 4, 5 can be rewritten as

$$\displaystyle{ \varDelta F^{{\ast}} = \frac{1} {2}(\eta _{m} -\eta _{f})(\mu -\mu _{\text{coex}})V ^{{\ast}}\quad. }$$
(7)

Thus the task of the simulation is to locate coexistence conditions ( p coex, μ coex, η m, η f) in the bulk, “measure” the chemical potential (or the pressure p l) of the liquid surrounding the nucleus, and “measure” the volume V∗ of the nucleus. The latter can simply be done by a finite-size generalization of the lever rule, when we carry out a “measurement” of p l in a simulation box of finite volume V box at a chosen constant packing fraction η

$$\displaystyle{ \eta V _{\text{box}} =\eta _{l}(\,p_{l})(V _{\text{box}} - V ^{{\ast}}) +\eta _{ c}(\,p_{c})V ^{{\ast}}\quad. }$$
(8)

As a caveat, we mention that V box has to be chosen large enough so that fluctuations of μ and p l are relatively small, and one can only work in a restricted range of packing fractions (avoiding both the “droplet-evaporation/condensation” transition [8] and the appearance of cylinder-like droplets or slab structures [9], see Fig. 3).

Fig. 3
figure 3

Schematic plot of the chemical potential μ vs. density ρ (or packing fraction η = (π σ c 3∕6)ρ, σ c being the colloid diameter, respectively), for a system undergoing a phase transition from liquid (at density ρ f) to solid (at density ρ m) at μ = μ coex in the thermodynamic limit (broken horizontal straight line) and in a box of finite volume V box (full curve). Due to finite size effects, the homogeneous liquid is stable until the density ρ 1 where the droplet evaporation/condensation transition occurs. For ρ 1 < ρ < ρ 2 a nucleus surrounded by liquid is stable: this is the region of interest, where μ l, p l, and V∗ need to be extracted. At ρ 2, a transition occurs to a cylinder-like nucleus, stabilized by the periodic boundary conditions that are applied throughout. For ρ = ρ 3 a transition to a slab-like crystal with two planar interfaces occurs. In the slab region, theory requires μ = μ coex, if the periodic boundary condition is commensurable with the crystal periodicity. The different states are illustrated with snapshot pictures of configurations of the model with η p r = 0. 1 (particles in the crystal are shown in red, in the fluid in blue, in the interface region in green) (From [6])

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Before presenting any details on our procedures, we show central results to show that the strategy outlined above works (Fig. 4). Indeed, apart from small deviations, for all choices of N used there is a broad regime where the proportionality of Δ F∗ to V∗2∕3 holds, and the important feature is that these data superimpose to a common straight line irrespective of N in each case. This property is crucial, because we want to be able to describe nucleation phenomena in bulk materials, not in nanoscopically small boxes with periodic boundary conditions. The use of such boxes is needed to be able to study nuclei in thermal equilibrium – a nucleus on top of the barrier in Fig. 1 is unstable against thermal fluctuations, of course, and cannot be straightforwardly studied.

Fig. 4
figure 4

Nucleation barriers Δ F∗(V∗) plotted vs V∗2∕3 for η p r = 0. 1 (a ) and η p r = 0. 2 (b ). Here units k B T = 1 and σ c = 1 are used. Three system sizes are included in each case, containing N = 6000, 8000 or 10, 000 particles, respectively. Full straight lines show fits assuming a spherical surface (replacing A w by A iso = (36π)1∕3 in Eq.(3)) and then fitting \(\bar{\gamma }\), with result \(\bar{\gamma }= 1.082\) (a ) and \(\bar{\gamma }= 2.406\) (b ). Thedotted lines indicate the predictions when one would take, in case (a ) γ 111 = 1. 013, γ 110 = 1. 044 and γ 100 = 1. 039, and in case (b ) γ 111 = 2. 078, γ 110 = 2. 224 and γ 100 = 2. 256 rather than \(\bar{\gamma }\). The inset in the figure shows the differences of the data to the fits (From [4])

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However, as the insets in Fig. 4a, b show, there occur minor deviations from the fit to a common straight line, but these deviations are of the order of a few percent only. These deviations are to a fewer extent statistical errors, but to a larger extent systematic. We attribute the systematic errors due to the fact that in our geometry the chemical potential in the system is not strictly constant (as tacitly assumed in Fig. 1), but fluctuates. This fluctuation is larger the smaller N is. A second systematic effect comes from the translational entropy of the nucleus in the simulation box, which scales proportional to ln(V box) and hence ln(N). More research will be needed to clarify the nature of these systematic corrections quantitatively.

In any case, the deviations due to the choice of a spherical droplet shape and use of any of the interface tensions of planar interfaces (γ 111, γ 110 or γ 100, respectively) are distinctly larger than these systematic errors, and would lead to a significant underestimation of the nucleation barrier, in particular for η p r = 0. 2. We expect that this discrepancy will increase further for still larger η p r, where ultimately faceted crystals [5] will result. We recall that in the simplistic lattice gas model at low temperatures T the nucleus shape tends to a simple cube, and then the ratio of the actual barrier Δ F∗ to the spherical approximation gradually tends to 6∕π as T → 0 [10].

In the next section, we shall give some details on the model that we have used for our study, and in the third section, some details of the actual analysis that yielded Fig. 4 will be given. Section 4 summarizes our conclusions, and gives an outlook to future work.

2 The Model and Its Bulk Properties

Our choice of model is motivated by colloidal suspensions, for which nucleation rates have been studied extensively both by experiment and simulations (see review [11]), focusing on the limit of hard-sphere like colloids. However, it must be noted that experimentally it is not possible to determine the packing fraction η better than with 1 % accuracy, moreover it is not possible to manufacture colloids which are perfectly uniform in size, and also some additional other interactions (in addition to the hard-sphere-like repulsion) are always present [12]. Hence we do not focus on hard spheres here, but rather focus on colloidal suspensions of hard-sphere-like particles where small polymers added to the suspension provide an attractive interaction, whose range can be controlled by the polymer radius and whose strength can be controlled by the polymer fugacity [13]. The standard model for such systems is the well-known Asakura-Oosawa (AO) model [14, 15]. Since the colloid-colloid interaction for this model is singular for distances equal to the colloid diameter σ c, it is inconvenient from a simulator’s perspective, and since it is also an idealization of reality anyway, it is advisable to work with the so called “softEffAO model” [4, 6, 7], where the singular potential is replaced by an almost equivalent smooth potential, which leads to a very similar phase separation as the original AO model. For r ≤ σ c( = 1) the repulsive potential hence is not infinite, but replaced by [k B T = 1]

$$\displaystyle{ U_{\mathrm{rep}}(r)\,=\,4\Big(\Big[ \frac{b\sigma _{c}} {r -\varepsilon \sigma _{c}}\Big]^{12}\,+\,\Big[ \frac{b\sigma _{c}} {r -\varepsilon \sigma _{c}}\Big]^{6}\,-\,\Big[ \frac{b\sigma _{c}} {b\sigma _{c}+q -\varepsilon \sigma _{c}}\Big]^{12}\,-\,\Big[ \frac{b\sigma _{c}} {b\sigma _{c} + q -\varepsilon \sigma _{c}}\Big]^{6}\Big)\;. }$$
(9)

where b = 0. 01, ε is specified below, q = 0. 15 is a convenient choice of constants. The attractive part of the potential is described by σ c < r ≤ σ c(1 + q)

$$\displaystyle{ U_{\mathrm{att}}(r) =\eta _{ p}^{r}\Big(1 + q^{-1}\Big)^{3}\Big[1 - \frac{3r/\sigma _{c}} {2(1 + q)} + \frac{(r/\sigma _{c})^{3}} {2(1 + q)^{3}}\Big]\quad, }$$
(10)

while U att(r > = σ c(1 + q)) = 0. The parameter ε is chosen such that the total potential is smoothly differentiable at r = σ c, which yields ε = 0. 967118 (η p r = 0. 1) or ε = 0. 9892 (η p r = 0. 2), respectively. For this potential it is straightforwardly possible to compute the pressure applying the Virial formula, unlike for the original AO model.

Figure 5 shows then the phase diagram of this model for different choices of η p r. These data were taken by sampling the packing fraction η by Monte Carlo runs in the NpT-ensemble, using N = 4000 colloidal particles. The data for the crystal were obtained using a perfect fcc lattice as initial condition, of course. Ideally, the liquid branch should only occur for pressures p ≤ p coex. However, as usually observed for NpT simulations of first order transitions, this is not the case: there is a regime of pressures where both phases are stable or metastable, respectively, and from the data of Fig. 5 a straightforward estimation of p coex is not possible. We have determined p coex by the method proposed by Zykova-Timan et al. [16]. In this method, one studies slab configurations where in the initial state a crystal domain (of volume V c = L × L × L c) and a liquid domain (of volume V l = L × L × (5L − L c)) are present. Periodic boundary conditions are used, and thus the domains are separated by two planar L × L interfaces (L is chosen such, that at the chosen pressure the crystal is not distorted). If the chosen pressure exceeds p coex and we let the system evolve in the Monte Carlo run, we expect that the crystal grows on expense of the liquid, while the opposite behavior occurs for p < p coex (see [16, 17] for more details). Plotting the volume change of the total system versus Monte Carlo “time” for various pressures we identify p coex as the pressure where no volume change occurs (Fig. 6). As discussed in [4, 7, 17], this method is not as straightforward as it looks, since there is both the need to take averages over many equivalent runs to reduce statistical noise in the curves such as shown in Fig. 6, and there is the need to study several choices of L (or n, respectively) to extrapolate the resulting estimates of p coex(n) vs n−2 in order to obtain an estimate for the true coexistence pressure that applies in the thermodynamic limit. When p coex is known and the liquid and solid branches η l( p) and η c( p) are known (Fig. 5), we immediately can read off η f = η l( p coex), η m = η c( p coex), and from the estimation of the pressure p l of the liquid coexisting with the nucleus we can infer μ {Eq. 6} from the linear expansion of p l {Eq.5}, and using also the expansion for p c {Eq. 5} we find η c( p c) and V∗ can then be inferred from Eq. 8. However, it is advisable to check that one works close enough to coexistence conditions such that the linear expansions, Eq 5, are actually valid. For this purpose, a new method to estimate the chemical potential has been developed [4, 6, 7, 17], since in many cases of interest the standard Widom particle insertion method [18] cannot be applied. Figure 7 shows, as an example, the chemical potential μ for η p r = 0. 2 plotted against both pressure and packing 

Fig. 5
figure 5

Normalized pressure p versus packing fraction η for several choices of attraction strength η p r, as indicated. The branches at the left side represent the liquid phase and the branches at the right side the crystal

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Fig. 6
figure 6

(a ) Volume change as a function of the number of Monte Carlo steps for η p r = 0. 2, choosing n = 10 lattice planes in x and y directions, and pressures from p = 0. 6 (red, top) to p = 3. 0 (magenta, bottom). (b ) Same plot as (a ) for η p r = 0. 28

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Fig. 7
figure 7

Chemical potential as a function of pressure (a ) and packing fraction (b ), for the softEffAO model with η p r = 0. 2

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fraction, using the estimate of p coex as estimated above from the method explained in Fig. 6. Indeed one finds that the curves μ vs. p for both phases are straight lines in the regime of interest. Using p coex = 1. 78 ± 0. 02 we found μ coex = −4. 60 ± 0. 04 in this case, leading to η f = 0. 374 ± 0. 002, η m = 0. 688 ± 0. 001.

3 Simulation Analysis of the Nucleus-Fluid Equilibrium

The nucleus-fluid equilibrium was studied by putting part of the particles in a subsystem with perfect crystal structure at η = η m and filling the rest of the box with particles having the expected density of the fluid, and then equilibrating the system. It is important to verify that the resulting crystal nucleus does not depend in any significant way on this rather arbitrary initial state. This fact is illustrated in Fig. 8, where it is illustrated for η p r = 0. 2 that different shapes of the initial crystal lead to rather similar shapes of the resulting equilibrated nucleus, and the corresponding distributions of pressures and density in the fluid region surrounding the nucleus are within error identical. Note that in this case p l ≈ 2. 454 and η l = 0. 4184 ± 0. 0001.

Fig. 8
figure 8

Different crystalline seeds (left column, part (a ), (b )) lead to very similar shapes of the equilibrated crystalline nuclei (next column, part (c ), (d )) and almost identical distributions of pressure (part (e ), (f )) and density (part (g ), (h )) of the surrounding fluid. All data refer to the case N = 10, 000, η = 0. 48, η p r = 0. 2. The equilibrated nuclei shapes where obtained after about 1010 Monte Carlo cycles, with each cycle comprised of N Monte Carlo trial moves

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The actual data for p l(η) for three different choices of N are then shown in Fig. 9. One can see that the variation of the pressure as function of the total packing fraction η decreases, in accordance with the schematic variation of μ vs. η in the region where the nucleus coexists with surrounding fluid (Fig. 3). The larger N the smaller the pressure (for N → ∞ these curves converge towards the horizontal variation p = p coex). From this study we find the packing fraction η l( p) of the surrounding fluid, at the same time, and these data are also shown in Fig. 9: they coincide at a common curve and this curve is nothing but the pressure vs. packing fraction curve of the corresponding homogeneous system (Fig. 5).

Fig. 9
figure 9

Pressure p l in the liquid surrounding a crystalline nucleus, as shown in Fig. 8, plotted vs. the total packing fraction η for the softEffAO model choosing η p r = 0. 2. Three choices of N are shown, N = 6000, 8000, and 10, 000, respectively. The region of interest is shown on the right with strongly magnified scales. Values for the packing fraction of the fluid η l are included and lie on top of the bulk equation of state for the liquid branch. From [4]

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Thus, our actual numerical results validate the assumptions made in our finite-size generalization of the lever rule, Eq. 8. From the knowledge of η, η l( p) and η c( p) (using Fig. 7) we have obtained all the necessary input to deduce V∗ via Eq. 8, and using then Eqs. 4 or Eq. 7, respectively, the data shown in Fig. 4 result.

4 Conclusions

A method to study the free energy barrier for homogeneous nucleation of crystals from a fluid phase has been presented, which is not hampered by the fact that the fluid-crystal interface tension in general is anisotropic. In the softEffAO model, variation of the parameter η p r that controls the strength of the effective attraction between the colloidal particles allows to control this anisotropy (Fig. 2), and indeed deviations from the standard (inappropriate) assumption of spherical nucleus shape were found (Fig. 4). In the present report, several steps of analysis of the simulation data have been explained. We also emphasize the need for accessing a fast supercomputer such as HORNET at the HLRS Stuttgart for the research: typical system sizes involve systems with 104 colloidal particles, and for obtaining data such as shown in Fig. 6 averages over 100 runs carried out in parallel need to be taken.

While the present work has addressed a simple model system, appropriate for colloidal suspensions, future work should address interparticle potentials that are relevant for materials science, since nucleation of crystals is a very relevant problem there. Also an application to study the formation of ice nuclei in the atmosphere is planned, since this problem is of central importance in the context of climate modeling. In all cases, complementary studies of the kinetic aspects of nucleation phenomena will be needed.