Heterogeneous nucleation of supercooled water, and the effect of an added catalyst

Abstract

The statistics of liquid-to-crystal nucleation are measured rigorously by using a recently developed automated lag-time apparatus (ALTA). A single sample, in this case a sample of pure water both with and without an (insoluble) AgI crystal, is repeatedly cooled, nucleated, and thawed. Analysis of the data, coupled with a second kind of experiment, shows that the statistics of nucleation are consistent with a first-order kinetic mechanism over a wide range of supercooling temperatures. The limitations of classical nucleation theory are exhibited. Our analysis unifies many related experiments in biology, physics, chemistry, and chemical engineering.

Liquid-to-solid nucleation of a supercooled aqueous solution is a simple but profound example of the poorly understood process of evolution of a metastable state to its final equilibrium state. Such processes are some of the least understood phenomena in biology (1–4), physics (5–9), chemistry (10–12), and engineering (13, 14). The lag-time before a supercooled sample nucleates to a solid is a stochastic function, strongly dependent on the degree of supercooling (15). Using a robust lag-time apparatus, and 300 repetitions on a single sample, we exhibit explicitly the statistics governing this lag-time distribution, which we show is consistent with a simple first-order kinetic model. Using a single assumption (subsequently tested), we extract the entire nucleation curve from a single linear supercooling experiment. Addition of a crystal of (insoluble) AgI simply shifts the entire nucleation statistics to higher temperatures. We show the equivalence of this phenomenon in widely separated fields of science: biological studies of the supercooling of fish blood (16) and insect hemolymph (17), the physics of alloy supercooling (18, 19), chemical engineering (13), and chemistry of supercooled aqueous solutions (20).

The data arising from our nucleation experiment are simple, as shown in Fig. 1. A single sample of pure water of volume 200 μl is placed in an NMR tube and cooled in our automated lag-time apparatus (ALTA 4) (20) below the freezing point of water until it nucleates heterogeneously. The temperature is decreased linearly by a feedback mechanism, in these experiments at the rate α = 1.08 K min⁻¹. The data collected are the lag-time τ until nucleation and the supercooled temperature ΔT at which nucleation occurs for each run, and they are plotted by the symbols in Fig. 1 Upper, both for the single sample pure water (circles, lower temperature of nucleation) and for the same water sample and NMR tube with a single crystal of (insoluble) AgI added (squares, higher temperatures of nucleation). The thin line is the actual recorded temperature of run 106 for pure water, which shows the linearity of our cooling. A total of 294 and 354 runs are collected for pure water without and with the AgI crystal added, and they show clearly the stochastic nature of nucleation. Elsewhere we show that these data for pure water can be collected reproducibly over more than an order of magnitude of cooling rates (20), but here we use the slowest cooling rate for simplicity.

Figure 1.

Automated lag-time apparatus (ALTA 4) data (Upper) and survival curves (Lower) for AgI (left; 354 repetitions) and pure water (right; 294 repetitions) heterogeneous nucleation at the linear cooling rate α = 1.08 K min⁻¹. The thin line in Upper is the actual recorded temperature of run 106 of the pure water experiment, to indicate the success of the feedback control.

We define the “survival curve” as the number of samples unfrozen after time t, N(t) divided by the total number of samples N₀. The survival curves are shown in Fig. 1 Lower, and they reveal crucial facts concerning nucleation, which have not been appreciated in some of the many fields in which versions of such experiments are performed. There is an inherent and reproducible width to each survival curve, which captures quantitatively the stochastic nature of heterogeneous nucleation. For our linear supercooling experiment, the horizontal axis may be viewed equivalently as either the time t the sample has existed below the equilibrium freezing point, or the supercooling temperature ΔT, since ΔT = α t. Our working definition of the so-called supercooling point (20), often approximated in biological studies, is the temperature at which, on average, 50% of the samples are frozen. Elsewhere we show this corresponds quite accurately to the maximum of the derivative of the survival curve, sometimes called the nucleation rate. The full width at half height of that derivative also corresponds approximately to the width of the (asymmetric) survival curve between 10% and 90% of the samples nucleated. Hence we recommend here that supercooling points be reported from the survival curve, and for our data these are 13.78 ± 1.4 K and 6.13 ± 1.3 K.

A remarkable feature of our data is that the survival curve for the pure water sample is identical to the survival curve measured after the AgI crystal is added to the tube, on the scale of the above figures, with the whole survival curve shifted by 7.65 K. There is a long and rich history of the effect of AgI on water freezing that extends back to Irving Langmuir (21, 22) and highly controversial cloud seeding experiments, which space does not permit us to discuss here. The water is EM Science OmniSolve reagent-grade water filtered through a 0.2-μm filter. When the solution contains nucleators, either insoluble particles such as the AgI crystal analyzed here, or apparent molecular nucleators such as in Antarctic fish blood and Norwegian insect hemolymph discussed elsewhere, the measured supercooling point is independent of which NMR tube is used. Once all of the nucleators are removed, for example in a pure water sample, we measure heterogeneous nucleation on a surface. Hence there can be a dependence of the supercooling point for pure water (but not other samples) on the glass surface. For this reason we identify which NMR tube is used in each experiment. All experiments presented here used the same NMR tube, coded 20010701 in our laboratory.

Much more can be extracted from our data with a single assumption, which is tested explicitly below. By analogy with macroscopic chemical kinetics, we assume that heterogeneous nucleation is a single-step kinetic process from “reactant” supercooled liquid A to “product” crystal B with rate constant k, which depends on the degree of supercooling ΔT. We have previously verified that this assumption holds for fixed supercooled experiments (15, 23), in which the sample temperature is lowered as quickly as possible to a fixed supercooled temperature, independent of time. In this case, the survival curve is given by

1

and this is verified explicitly below. The average lag-time is 〈τ〉 = k⁻¹. The thermal histories for the two complementary modes of operation of our automated lag-time apparatus are shown in Fig. 2.

Figure 2.

The thermal history of nucleation experiments for AgI-seeded pure water samples. (Upper) Actual recorded temperature as a function of time runs 15 and 16 in linear supercooling mode, at linear cooling rate α = 1.08 K min⁻¹. (Lower) Runs 72 and 73 in the constant supercooled temperature mode.

Our analysis of the linear supercooling data is concluded by assuming a functional form for the nucleation rate constant k as a function of supercooling temperature ΔT. We call this crucial function the nucleation curve. The limits of this function are clear. At small supercooled temperatures, the metastable liquid survives on average for extremely long times, because there is no “driving force” for nucleation, and the average lag-time 〈τ〉 is extremely large. For deeply supercooled samples, the lag-time is small. However, there is no theory at present to explain the full range of supercooling temperatures, and the explanation of our data is a test that a candidate theory of liquid-to-solid nucleation must pass. Here, without favoring any particular microscopic theory for this function, in our first analysis we adopt the functional form (but not microscopic parameters) of classical nucleation theory, which yields for this function

2

where the constants P and Wb³ are approximated by a collection of material parameters in classical nucleation theory, but here simply fit to the survival curves in Fig. 1, yielding the fitted values for pure water P = 1.78 × 10⁻¹⁰ sec⁻¹ and W^1/3b = 22.34 K, and with AgI crystal added P = 2.12 × 10⁻⁴ sec⁻¹ and W^1/3b = 29.70 K.

With this one assumption, the average lag-time as a function of supercooling temperature may be extracted by nonlinear fitting from our survival curves, and this is exhibited in Fig. 3. We believe this is the first time that this full curve has been extracted directly from a single experiment. Note the extremely steep decrease in average lag-time as deeper supercooling temperatures are reached. Again the pure water curve is superposable on the AgI curve with a shift of 7.65 K. Note that at extreme supercoolings, where a glassy phase may form, one might expect the nucleation curve to again turn upward as sampling dynamics rather than free energy barrier crossing dominates the nucleation event. We have not yet seen that phenomenon with ALTA 4.

Figure 3.

The average lag-time 〈τ〉 as a function of supercooling temperature ΔT for AgI-seeded water (left) and pure water (right). The AgI curve is displaced exactly 7.65 K from the pure water curve, from which it is indistinguishable on this scale when so displaced.

For the AgI sample, one may read off the predicted survival curves for fixed supercooled temperatures of 4.8 K, 5.1 K, and 5.6 K, and these are shown in Fig. 4. The extreme sensitivity of the average life-time to supercooling temperature is manifest.

Figure 4.

Explicit test of the only assumption in the deconvolution of the ALTA 4 data. The solid lines are the predicted survival curves extracted from the nucleation curve of Fig. 3 for fixed supercooled temperatures 4.8 K, 5.1 K, and 5.6 K, respectively. The points are the survival curve actual data collected in a constant supercooled temperature mode experiment on the same sample, at the supercooled temperature 4.9 ± 0.1 K.

Our experiment is completed by running our apparatus in the fixed supercooled temperature mode, on exactly the same AgI-seeded pure water sample, cooling rapidly to 4.9 ± 0.1 K below the freezing point of pure water and waiting until the sample nucleates. The results of 248 repetitions of this experiment are plotted as the symbols in Fig. 4. Within statistical accuracy, the assumption described in the text above is verified, and the loop of consistent analysis is closed.

Our experiment unifies the discussion of nucleation experiments in at least five fields of science and engineering: gas clathrate hydrates (10, 13), ice chemistry and physics (5), cold adaptation in biology (3, 4, 16, 24), alloy metallurgy (19), and nucleation by sonification (14, 25). Unfortunately, in some fields a tradition has arisen of analyzing nucleation data with histograms having arbitrarily selected bin widths. The use of survival curves as shown above avoids such arbitrary analysis (23, 26). To show the unity of these fields, we have digitized some histogram data from the alloy (19) and sonification (14) literature, and have reconstructed (as accurately as the published data will allow) the survival curves, and these are shown in Fig. 5.

Figure 5.

Nucleation survival curves from various branches of science. From left to right, the AgI-seeded pure water sample discussed here (dashed line), a 2% solution of the hemolymph of the Norwegian bark beetle Rhagium inquisitor, solution of blood serum from the New Zealand black cod Notothenia angustata, solution of blood serum from the Antarctica fish Pagothenia borchgrevinki, digitized data for sonificated water (14) (dot-dashed line), pure water results from this paper (dashed line), and digitized supercooling data for cobalt/palladium alloys at four concentration ratios, 82:18, 75:25, 70:30, and 50:50 from Schenk et al. (19).

Our experiments have much to say about classical nucleation theory, developed originally for gas-to-liquid nucleation, where there is a massive density change, and often extended to try to cover the liquid-to-crystal case. This theory predicts that homogeneous nucleation of water occurs around −40°C, with a critical cluster size of approximately 3,000 molecules, much larger than nonclassical theories suggest. Heterogeneous nucleation is often approximated by using a so-called catalytic potency factor (27) W = W(θ), which appears in the exponent of Eq, 2 above. This correction factor contains a contact angle θ for the nucleating liquid on a substrate,

3

Using a standard value for the (assumed curvature-independent) ice/water surface free energy of 30 mJ m⁻², our data may be used to extract values for the factor W(θ), and hence contact angle values θ_water = 166.54° and θ_AgI = 163.3°. These are extremely steep contact angles, and lead to tiny values for W(θ). Interestingly, they are similar to contact angles reported in a study of water drops supporting aliphatic alcohol Langmuir films, which found (with above definition of contact angle) 161° and 162° for pentacosanol (C₂₅H₅₂O) and heptacosanol (C₂₇H₅₆O) films, respectively (9). The prefactor for pure water extracted above is comparable to some values cited for homogeneous nucleation of 10^32±1 cm⁻³ sec⁻¹ (27) and 10^33±3 cm⁻³ sec⁻¹ (26).

We have shown that approximately 300 repetitions of an identical linear supercooling experiment, together with a verified approximation, yield the entire nucleation curve in a single experiment. Adding a single (insoluble) AgI crystal to a sample of pure water shifts this entire survival curve to warmer temperatures by 7.65 K. These data provide a rigorous test for candidate theories of liquid-to-solid nucleation, and a sound and reproducible basis for reporting supercooling points in a variety of scientific literature. The challenge to theory remains to explain both the natural width of the survival curves (or equivalently the nucleation probability), and the molecular origin of the shift for nucleators such as AgI.

Abbreviation

ALTA 4

automated lag-time apparatus