Solution Interface Studied

CRYSTAL GROWTH & DESIGN

Freeze Concentration of Solutes at the Ice/Solution Interface Studied by Optical Interferometry

2002 VOL. 2, NO. 6 541-548

Michael F. Butler* Unilever Research and Development, Colworth House, Sharnbrook, Bedfordshire, MK44 1LQ, United Kingdom Received September 30, 2002

ABSTRACT: Mach-Zender optical interferometry was used to measure the solute concentration profile in solution near the ice/solution interface under quasi-steady state conditions in order to gain a quantitative understanding of freeze concentration. At low ice growth velocities, the freeze concentration behavior of a range of solutes, including sucrose, a globular protein, and some low molecular weight polymers, was accurately described by the quasi-steady state approximation to the diffusion equation. Increasing deviations from this approximation occurred at higher growth speeds and for larger macromolecules, where diffusion was presumably inhibited by chain entanglement that led to the development of significant concentration gradients parallel to the ice/solution interface. In mixtures of noninteracting solutes, the concentration profiles of the individual components could be distinguished separately. The behavior of a helical antifreeze glycopeptide could not be explained in terms of its diffusion behavior, however. Its equal amount of incorporation into the ice phase at all growth speeds led to the conclusion that it interacted with the ice interface by virtue of possessing significant crystallographic similarities with the ice lattice. The description of freeze concentration at the ice/solution interface by the quasi-steady state approximation is therefore limited to low molecular weight (nonentangled) solutes, low growth speeds, and the absence of substantial interactions between the ice phase and the solute. Introduction It is well-known that redistribution of solute occurs during crystallization of most materials, including ice,1-6 to an extent that is predominantly determined by the solubility of the solute in the solid phase and the ability of the solute to diffuse away from the interface before it is overgrown. At very low crystal growth speeds, the solute is expected to diffuse away from the growing crystal (unless it possesses crystallographic similarities with the growing crystal that favor adsorption) leading to a pure crystal and highly enriched solution (the case of no solute incorporation). At high crystal growth speeds, the solute is unable to diffuse away from the growing crystal and is completely engulfed, leading to an impure crystal and no enrichment of the solution (the case of complete solute incorporation). At intermediate crystal growth speeds, which cover many practical circumstances, some solute is incorporated into the crystal and some is rejected, leading to enrichment of the solution phase (the case of partial solute incorporation). To understand the solute concentration profiles that form in the vicinity of the solid/liquid interface in the general case of partial solute incorporation, it is necessary to consider the diffusion equation that describes the movement of solute molecules in relation to the advancing crystal growth front. For a one-dimensional system, the general solute conservation equation for the liquid in a constant velocity moving coordinate frame, where ξ is the distance from the interface, is given by1

{

[

] }

∂CS ˜0 D ∂δ∆G ∂ D∂CS ) + V+ - u(ξ) CS ∂t ∂ξ ∂ξ κT ∂ξ

(1)

* To whom correspondence should be addressed. Tel: +44 (0)1234 222958.Fax: +44(0)1234222757.E-mail: [email protected].

The first term represents Fickian diffusion. CS represents the solute concentration in the solution, and D is the solute diffusion coefficient that accounts for the general mobility of the solute molecules owing to their size, shape, and possible interactions with their neighbors. The second term accounts for the moving interface, with V being the growth velocity. The third and fourth terms are field terms. The former is due to internal interfacial effects caused by, for example, the charge present at the ice/solution interface and corresponding solute conformational changes, and the latter is due to external fields, for example, an applied electric or magnetic field. The chemical potential of a molecule can be quite different in an interface region as compared to the bulk phases due to local electrostatic, magnetic, or strain effects. Because thermodynamic equilibrium demands a constant value for the extended electrochemical potential, a redistribution of the solute atoms or molecules must occur in the interface region in response to these field effects. Therefore, it is possible for a solute concentration gradient to become established even at a stationary interface in the presence of an adsorbing species. In the ice/water system, the different partitioning in the ice of the dissociated H+ and OH- ions is responsible for an interface field, which may cause a variation in solute concentration that extends for about 0.1-1.0 µm into the solution.2 Assuming no diffusion within the solid phase and ignoring the field terms in the diffusion equation, an approximate steady state solution (the quasi-steady state approximation) can be formulated,1,3 which is depicted schematically in Figure 1.

CS(ξ) Cs(∞)

)1+

(1 - k) Vξ exp k D

(

)

(2)

where k is the effective partition coefficient that neglecting solute diffusion within the solid, is defined as

10.1021/cg025591e CCC: $22.00 © 2002 American Chemical Society Published on Web 10/12/2002

542 Crystal Growth & Design, Vol. 2, No. 6, 2002

Butler Table 1. Values for the Diffusion Coefficient of All the Samples Studied sample

D (µm2 s-1)

sample

D (µm2 s-1)

sucrose pullulan BSA

210 24 33

AFGP PVA 9K PEG

116 29 85

is used by beverage manufacturers to form fruit juice concentrates. Experimental Section

Figure 1. Distribution of solute in the solution during the build up of solute toward the quasi-steady state distribution, CS. CT shows the solute concentration at the interface, and CC is the solute concentration in the solid phase (after ref 3).

the ratio of the concentration of solute that is trapped in the solid, Cc(0) to the bulk concentration in the solution, Cs(∞).1-3 This solution is only valid when the solute concentration in the enriched phase changes slowly. In reality, the solution becomes progressively more enriched so that a genuine steady state is not reached, unless liquid inclusions are formed inside the solid phase, and there is a gradual increase in the value of Cs(∞). However, if the volume of solution is sufficiently large, then the increase in Cs(∞) over the course of the experiment can be considered to be negligible, the quasi-steady state conditions are relevant, and eq 2 is valid. In the quasi-steady state approximation, the time dependence of the solute concentration at the interface, CT, as a function of the absolute distance from the initial position of the interface, leading up to the steady state value, is described by3

CT(x,t) Cs(∞)

)

{1 -k k [1 - exp(- kRD)x] + 1}

(3)

From eqs 2 and 3, conservation of solute leads to the following expression for the steady state solute distribution in the solid, CC:

CC(x) Cs(∞)

{

[

(

) (1 - k) 1 - exp -

)] }

kVx +k D

(4)

The functions represented by eqs 2-4 are shown schematically in Figure 1. The aim of the work presented in this paper was to relate the solute concentration profile in the solution ahead of the ice/solution interface, for a number of solutes, to the solute transport properties and potential interfacial activity via the solutions to the diffusion equation presented earlier, grown under quasi-steady state conditions. As a sensitive probe of the spatial refractive index (and hence concentration) variation on microscopic length scales, a Mach-Zender optical interferometer was used to measure the concentration profiles of the solute in solution. This type of study not only is of academic interest, as a study of the factors influencing ice growth in solution, but also has technological significance.4-6 For example, freeze concentration

Materials, Sample Preparation, and Sample Characterization. The following materials were used in the study presented in this report: sucrose, pullulan (Mw ∼ 480 000 from Sigma Aldrich), antifreeze glycoprotein (AFGP, Research Grade from AF Protein Inc.), bovine serum albumin (BSA, from Sigma Aldrich), poly(vinyl alcohol) (PVA, Mw 9000, from Sigma Aldrich), and poly(ethylene glycol) (PEG, Mw 8000, from Sigma Aldrich). Solutions (1 wt %) were made from all of the materials, using distilled, deionized water, with the exception of sucrose, from which a 5 wt % solution was made. In addition, a mixture with a final composition of 5 wt % sucrose and 1 wt % pullulan was made. Solutions of 0.25 and 0.5 wt % (1.25 and 2.5 wt % for sucrose) were made by successively diluting the 1 wt % solutions for the measurement of the concentration dependence of the refractive index. Refractive index measurements were performed over a range of temperatures above 20 °C using a refractometer operating at a wavelength of 532 nm. A linear relation was found that was extrapolated back to 0 °C. The refractive index at this temperature was then used to construct the linear relationship between refractive index and solution concentration. The diffusion coefficient, D, of the solutes at their initial concentrations was measured using dynamic light scattering, performed with a Malvern Instruments 4700 Autosizer. The correlation function was measured at a scattering angle of 90° for a range of temperatures between 8 and 30 °C and was accurately described in all cases by a single exponential decay. The diffusion coefficient was extracted from the correlation function, assuming the molecules occupied a spherical volume, at each temperature and formed a linear relationship that was extrapolated back to 0 °C. The values of the diffusion coefficient thereby obtained at 0 °C are shown in Table 1. Interferometry. The interferometer that was constructed for the purpose of measuring the interface morphology and the solute concentration field ahead of ice/solution interface was of the Mach-Zender type and has been described previously.7 It was chosen because of its ability to simultaneously image the interference fringes, from which the concentration field was calculated, and the ice/solution interface. The concentration profiles were extracted from the deviation of the interference fringes from a reference position by the following expression5

CS - C0 λ∆x ) CS (n(C) - n0)hw

(5)

where CS is the local solution concentration, C0 is the bulk concentration, λ is the laser wavelength (632.8 nm), ∆x is the deviation of the fringe position from that of the reference fringe, n(C) is the refractive index of the solution with concentration C, n0 is the refractive index of pure water (1.3327), h is the sample thickness (75 µm), and w is the separation of the reference fringes. Directional Ice Growth. The ice structures that were examined using optical interferometry were grown in a temperature gradient microscope stage (Linkam Scientific Instruments model GS150), which consisted of two silver plates held at different temperatures and separated by a gap of 4 mm. The rectangular glass coverslip, of thickness 0.17 mm, on which the sample was placed, was positioned across the gap

Freeze Concentration Studied by Optical Interferometry

Figure 2. Interference fringe profile in a BSA solution grown with a planar ice/solution interface at 2 µm s-1 (a) before initiation of ice growth and (b) during quasi-steady state growth, showing similar fringe curvature in the ice phase at all times. between two plates so that a temperature gradient existed across it. The slide was held in place by a movable sprung plate connected to a motor capable of moving the sample across the gap at speeds between 0.1 and 6000 µm/s. The sample itself consisted of a drop of solution surrounded by a 75 µm spacer on top of which a standard microscope coverslip (thickness 0.17 mm, diameter 22 mm) was placed. The silver blocks on which the sample was placed were cooled using liquid nitrogen to achieve temperatures (monitored by thermocouples inside the plates) of -15 and +15 °C on the “cold” and “hot” plates of the stage, respectively. The sample environment was sealed and purged with dry nitrogen gas to prevent condensation forming on the top and bottom surfaces of the sample. Once formed, the ice solution interface stabilized and became uniform and planar within a few minutes. Ice growth was initiated by activating the motor and moving the sample toward the cold plate, thus cooling the solution ahead of the ice solution interface. Ice was grown at speeds of 2, 5, 10, and 20 µm s-1.

Results Solute Concentration Profiles in Solution (Single Solute Systems). In all cases, the ice/solution interface remained planar for the growth speed of 2 µm s-1. At higher growth speeds, the interface became unstable and a cellular or dendritic morphology developed. Detailed analysis of the results was therefore confined to the slowest growth rate, at which the interface was planar. No concentration gradients were measured in any sample when the ice/solution interface remained static. Figure 2 shows interference images obtained at the start of the experiment and at the quasi-steady state for BSA, as a representative example. Although there was some initial curvature in the ice phase, it remained constant during planar growth. At the growth speeds for which the interface became unstable, shown in Figure 3 for sucrose as a representative example, and developed into a cellular or dendritic morphology, the curvature of the interface increased prior to the development of the instability, however. Grain boundaries, marked “gb” in Figures 2 and 3, were present in the ice phase in all of the samples. The interference fringes in the cases where a planar interface propagated were uniform and parallel inside the ice phase, indicating no change in refractive index, hence ice thickness, and

Crystal Growth & Design, Vol. 2, No. 6, 2002 543

Figure 3. Increase in curvature of the interference fringes in the ice phase with time for ice grown in a sucrose solution at 5 µm s-1, when the interface became unstable and developed a cellular morphology. (a) Before growth, (b and c) during development of interfacial instabilities, (d) final cellular morphology.

there were no signs of liquid inclusions. In the case where the interface became cellular or dendritic, the fringes became very distorted in the ice phase, as shown in Figure 3d, and liquid inclusions were present in the regions between the cells and the dendrites. Figure 4 shows the development of the concentration profile and the interfacial concentration for the samples that yielded complete agreement with the predicted quasi-steady state concentration profiles, i.e., assuming that the bulk solution concentration change is very slow over the course of the experiment. The solid line in the plots of the interfacial concentration is the prediction of the value of the interfacial concentration from eq 3. The solid line in the concentration profiles is a single exponential fit that is the functional form expected for the quasi-steady state concentration profile (eq 2). In order of descending diffusion coefficient, the solutes that exhibited agreement with the quasi-steady state prediction were sucrose, PEG, BSA, and PVA. These samples also did not develop any concentration gradients parallel to the ice/solution interface, although grain boundaries were present in the ice phase in all cases. The solute concentration profiles for pullulan and AFGP did not behave according to the predicted profiles, however. For pullulan, significant concentration gradients parallel to the ice/solution interface became established, shown in Figure 5a by the greater degree of fringe bending in the vicinity of a grain boundary, and the concentration distribution was not therefore as described by the simple quasi-steady state model. The corresponding concentration profiles at different positions along the interface are shown in Figure 5b. The development of the solute concentration profile for pullulan at a position well away from a grain boundary is shown in Figure 6a, and the predicted and measured values of the interfacial concentration are shown in Figure 6b. Although the concentration profile was reasonably well-described by an exponential decay, the amount of pullulan at the interface did not accumulate as predicted by the expression given earlier. AFGP displayed completely different behavior from any of the other solutes, with no concentration gradient developing at any time, at any growth speed (shown in Figure 7).

544

Crystal Growth & Design, Vol. 2, No. 6, 2002

Butler

Figure 4. Development of the solute concentration profile in solution (i) and interfacial concentration (ii) for (a) sucrose, (b) PEG, (c) BSA, and (d) PVA.


Figure 5. (a) Interference image showing the lateral variation of solute concentration near the interface in the pullulan sample. The ice/solution interface and grain boundary within the ice are shown by the dotted white line. (b) Solute concentration profiles at the three positions marked in panel a.

Table 2 shows a comparison between the value of the quasi-steady state V/D ratio that was measured from the fit of eq 2 to the measured concentration profile and the values that were predicted from the applied growth speed of 2 µm s-1 and known diffusion coeffcient for each sample. The measured and predicted values were in agreement for the solutes whose solute concentration profile and interfacial concentration agreed with the values predicted by the quasi-steady state approximation. No agreement was obtained for solutes whose concentration profiles deviated from a single exponential decay or were not described by the steady state approximation. Figure 8 shows the quasi-steady state concentration profile for sucrose at growth speeds of 2, 5, and 10 µm s-1, as a typical example. At speeds of 5 and 10 µm s-1, the growth front was dendritic rather than planar and a significant amount of solution was trapped between the dendrites as liquid inclusions. In the cases where a cellular or dendritic morphology developed, the interfacial position was taken to be the locus of the cell or dendrite tips. For all of the solutes (except AFGP), the amount of solute that accumulated at the interface decreased with increasing growth speed. For a given


Figure 6. (a) Development of the concentration profile for 1 wt % pullulan grown at 2 µm s-1. The red symbols denote the steady state concentration profile. (b) Development of the interfacial concentration toward the steady state value and comparison with the theoretical values (solid line).

Figure 7. Concentration profile at all times for AFGP.

growth speed, the extent of accumulation decreased with decreasing solute diffusion coefficient. No solute accumulation was measured at all for pullulan solutions at growth speeds greater than 10 µm s-1, however. No solute accumulation was measured at any growth speed for AFGP. Although the concentration profiles were still reasonably well-described by a single exponential decay for sucrose, BSA, PEG, and PVA solutions at growth speeds where a cellular or dendritic front developed, the measured V/D values deviated from the predicted ones

546


Butler

Table 2. Comparison between the Measured and the Predicted Shape of the Steady State Concentration Profile for All Samples Grown at 2 µm s-1a sample

V/D measured (5%

V/D predicted (1%

sucrose BSA PVA PEG pullulan AFGP

0.01 0.06 0.07 0.02 0.06 0

0.01 0.06 0.07 0.02 0.08 0.01

a The uncertainty (expressed as a percentage) for the measured values was obtained from five repeat measurements of the concentration profile, and for the predicted values, it was obtained from three repeat measurements of the diffusion coefficient.

Figure 8. Effect of growth speed on solute accumulation for a 5 wt % sucrose solution, as a typical example. Table 3. Comparison between the V/D Ratio Measured from the Concentration Profiles (Left Column), Maximum Uncertainty 5%, and the V/D Ratio Predicted from the Applied Growth Velocity and Solute Diffusion Coefficients (Right Column); Uncertainty 1% for Different Growth Speeds sample sucrose PEG BSA PVA (Mw 9000)

2 µm s-1 0.01 0.02 0.06 0.07

0.01 0.02 0.06 0.07

5 µm s-1 0.02 0.06 0.04 0.10

0.02 0.06 0.15 0.17

10 µm s-1 0.03 0.10 0.17 0.18

20 µm s-1

0.05 0.12 0.30 0.24 0.61 0.34

to an extent that increased with increasing growth speed. For a given growth speed, the extent of the deviation increased with decreasing solute diffusion coefficient (Table 3). Solute Concentration Profiles Solutions (Mixed Solute System). The measured fringe deviation from the reference position in a mixture containing 5 wt % sucrose and 1 wt % pullulan grown at 2 µm s-1 at an interfacial position well away from any grain boundaries is shown in Figure 9a. The interface remained planar in this case. These solutes were chosen because the concentration profiles of the individual components were described well by single exponential decays but with a decay length that was different by about an order of magnitude. The fringe deviation in the mixture was indeed described more accurately by two exponential decay curves rather than the single exponential found for the solutions containing only one solute species. The broken lines show these two exponential decay curves, and the overall fitted curve is the solid black line. The exponential decay with the long decay length was labeled as the “sucrose” component, and the one with

Figure 9. (a) Fringe deviation from the reference position for a mixture of 5 wt % sucrose and 1 wt % pullulan shown with a fitted curve consisting of two exponential decays. (b) The calculated concentration profile from the sucrose component (solid line), along with the data measured from a pure 5 wt % sucrose solution after the same time from the onset of growth as the profile in Figure 8a was measured. (c) The calculated concentration profile from the pullulan component (solid line), along with the data measured from a pure 1 wt % sucrose solution after the same time from the onset of growth as the profile in Figure 8a was measured. In all cases, the interface was planar.

the short decay length was labeled as the “pullulan” component. The symbols in Figure 8b show the concentration profile near a planar interface in a pure 5 wt % sucrose sample measured at the same time after growth was initiated as the data was measured in the mixed system that is shown in Figure 9a. The solid line in Figure 9b represents the concentration of sucrose calculated from the fringe deviation of the sucrose component in Figure 9a. Likewise, the symbols in Figure 9c represent the concentration profile near a planar interface in a pure 1 wt % pullulan and the solid line in Figure 9c represents the concentration calculated from the fringe deviation of the pullulan component in Figure


9a. Perfect agreement was obtained for both the sucrose and the pullulan components in the mixed system with that expected from the individual components in the pure systems containing only a single component. Discussion It is important first of all to establish the validity of the analysis, which depends on the assumption of twodimensionality of the sample. Experiments on ice growth in potassium chloride solutions exist that suggest that in some cases even thin sample cells can exhibit a significant degree of three-dimensional phenomena such as convection.8 In the potassium chloride system, during experiments on dendrite growth, the interference fringes were curved near the ice/solution interface, which suggested a curvature of that interface.8-10 Likewise, the interfaces were also slightly curved, but to a lesser extent, in the current experiments. In the potassium chloride experiments, however, a large increase in the interfacial curvature was measured during growth that coincided with the onset of interfacial instability, leading to the growth of several layers of dendrites throughout the sample and the formation of a concentration gradient parallel to the sample thickness.8 A similar effect may have occurred in the present experiment for the growth conditions for which a cellular or dendritic morphology developed. However, in the situation where the growth front remained planar, it is believed that even if a concentration gradient did build up parallel to the sample thickness, it was not sufficiently large to cause a significant deviation from two-dimensionality in the sample for the following reasons. First, the current experiments were performed in a sample cell that was at least half as thin as the sample cell used in the potassium chloride experiments and was therefore less prone to convective effects that could lead to the formation of a concentration gradient parallel to the sample thickness. In addition, when dendrites were formed in the current experiment, only one layer of dendrites was observed. In the potassium chloride experiments, at least three layers were observed when the sample was viewed across its thickness.8 These observations suggest that conditions were closer to twodimensionality in the present experiment than in the potassium chloride experiments where the concentration gradient parallel to the sample thickness was measured. Second, when the interface remained planar in the current experiment, there was no significant change in the interfacial curvature during growth. In the potassium chloride experiments, the interface had become significantly nonplanar and appreciably more curved at the time at which a significant concentration gradient parallel to the interface had developed. When the interface was still planar in those experiments, the concentration was relatively uniform throughout the sample thickness. Third, because the potassium chloride experiments were designed to test theories of interfacial instability, they were performed under conditions where the interfacial solute concentration increased more rapidly and convective effects were more likely to occur. While the potassium chloride experiments showed that in some cases even relatively thin growth cells cannot be regarded as being truly two-dimensional, the agree-


ment between the experimentally measured solute concentration profiles in certain cases and the predictions of the quasi-steady state theory is believed to be genuine and not simply coincidental. Assuming two-dimensionality of the sample, therefore, with the exception of AFGP, the behavior of the solutes at the interface can be explained in terms of their bulk diffusion properties. “Small” molecules that diffuse freely and independently of each other behave as dictated by the diffusion equation at low growth speeds, at least when the ice/solution interface remains planar. Previous work on ice growth in solutions containing sugars and potassium chloride has shown that dendrite growth in these systems could also be explained entirely in terms of the diffusion characteristics of the solutes.7-11 Larger macromolecules such as pullulan are likely to possess more complex diffusion properties because, at the concentrations used, they do not diffuse freely. Instead, they are entangled and can only diffuse by reptation, which is motion along their length within a confined volume.12 Their slow diffusion rate causes them to be overgrown at lower ice growth speeds than for the smaller molecules. It also prevents the dispersion of the lateral solute accumulation at the grain boundaries. For pullulan, the diffusion rate was so low that the ice growth rate exceeded the solute relaxation rate. The inequality tS . R2/D, where ts is the time taken for the concentration profile to develop and R is the accumulated solute layer thickness, necessary for the quasisteady state conditions that allow the theoretical expression for the quasi-steady state concentration profile to be used,1 was not valid. The expressions for the quasisteady state concentration profile and development of the interfacial concentration were therefore not valid. The disparity between the measured concentration profiles and those predicted from the quasi-steady state theory where growth was cellular or dendritic is unsurprising since the concentration profile in these situations is predicted by different sets of equations.1 Although they have been successfully tested for the growth of isolated dendrites in an isothermal solution,13-15 the prediction for an array of overlapping dendrites in a temperature gradient is more difficult and will have to account for lateral diffusion of solute into the gaps between the cells or the dendrites and the formation of liquid inclusions between them.1 Such effects will be more significant for larger, slower molecules and will therefore occur at lower growth speeds for the molecules with smaller diffusion coefficients, as observed. Increased concentrations of sucrose between primary dendrite branches have been observed previously.7 It should also be noted that at higher crystal growth velocities interfacial adsorption can lead to a dependence of the partition coefficient on the crystallographic orientation.2 Because the crystalline anisotropy between the basal and the edge planes is important in determining the substantially different tip radii in ice dendrites, it is possible that solute partitioning is different in the cases of planar and dendritic growth. In any case, the concentration profiles for the dendritic and planar growth morphologies will certainly differ. Bulk diffusion is not sufficient to explain the behavior of AFGP, however. AFGP is believed to actively adsorb

548


to the ice surface via hydrophilic side groups that possess the same spacing as the water molecules parallel to the a-axis (the [101 h 0] plane) of ice.16-18 The lack of a concentration gradient at the interface for AFGP was therefore because surface effects rather than bulk diffusion controlled ice crystallization in this case. Furthermore, the partition coefficient of unity suggested by these results is consistent with there being a significant crystallographic similarity between the AFGP molecules and the ice phase. The absence of a measurable concentration profile at a static interface suggests, however, that the interface field expected to be established by an adsorbing species such as AFGP extended over distances lower than the optical resolution of the interferometer (∼a few micrometers). It should be noted that PVA has been suggested to be interfacially active at the ice surface.19 The explanation of the present results for PVA entirely in terms of diffusion suggests the importance of the length and time scale required for interfacial activity to become established as well as the strength of the interfacial interaction. The length scale set by diffusion was of the order of tens of micrometers whereas the length scale set by adsorption was less than a few micrometers (estimated, as for AFGP, by the absence of a measurable concentration gradient at a static interface). If the interaction between PVA and the ice surface was not particularly strong, then it is likely that the rate of ice growth exceeded the rate at which interfacial activity could become influential; therefore, diffusion was the dominant process that determined the concentration profile in this case. Conclusions Mach-Zender optical interferometry provided an accurate measurement of the solute concentration field ahead of a planar interface and allowed theories that predicted the concentration variation ahead of an interface to be tested. With the exception of AFGP, the freeze concentration behavior of small solutes, such as sucrose, globular proteins, and low molecular weight polymers, behaved as expected from considerations of bulk diffusion alone. At low growth velocities, the concentration profiles could be explained in terms of the quasi-steady state approximation to the diffusion equation, although there was an increasing deviation from this description at higher growth velocities. The larger

Butler

the molecule, and hence the lower its diffusion coefficient, the lower the growth velocity at which freeze concentration profiles could not be explained by the quasi-steady state approximation. In mixtures of noninteracting solutes, the concentration profiles of the individual components could be distinguished separately. The behavior of AFGP could not be explained in terms of its diffusion behavior. Its equal amount of incorporation into the ice phase at all growth speeds led to the conclusion that it interacted with the ice interface by virtue of possessing significant crystallographic similarities with the ice lattice. The description of freeze concentration at the ice/solution interface by the quasi-steady state approximation is therefore limited to low molecular weight (nonentangled) solutes, low growth speeds, and the absence of substantial interactions between the ice phase and the solute. References (1) Tiller, W. A. The Science of Crystallization: Macroscopic Phenomena and Defect Generation; Cambridge University Press: Cambridge, 1991. (2) Tiller, W. A. The Science of Crystallization: Microscopic Interfacial Phenomena; Cambridge University Press: Cambridge, 1991. (3) Tiller, W. A.; Jackson, K. A.; Rutter, J. W.; Chalmers, B. Acta Metall. 1953, 1, 428. (4) Flesland, O. Drying Technol. 1995, 13, 1713. (5) Chen, P.; Chen, X. D.; Free, K. W. J. Food. Eng. 1998, 38, 1. (6) Chen, P.; Chen, X. D. Can. J. Chem. Eng. 2000, 78, 312. (7) Butler, M. F. Cryst. Growth Des. 2001, 1, 213. (8) Nagashima, K.; Furukawa, Y. Physica D 2000, 147, 177. (9) Nagashima, K.; Furukawa, Y. J. Phys. Chem. 1997, 101, 6174. (10) Nagashima, K.; Furukawa, Y. J. Cryst. Growth 2000, 209, 167. (11) Butler, M. F. Cryst. Growth Des. 2002, 2, 59. (12) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986; pp 188-288. (13) Kostianovski, S.; Lipson, S. G.; Ribak, E. N. Appl. Opt. 1993, 32, 4744. (14) Notcovich, A. G.; Lipson, S. G. Physica A 1998, 257, 454. (15) Emsellem, V.; Tabeling, P. Europhys. Lett. 1994, 25, 277. (16) Brown, R. A.; Yeh, Y.; Burcham, T. S.; Feeney, R. E. Biopolymers 1985, 24, 1265. (17) Burcham, T. S.; Osuga, D. T.; Yeh, Y.; Feeney, R. E. J. Biol. Chem. 1986, 261, 6390. (18) Harrison, K.; Hallett, J.; Burcham, T. S.; Feeney, R. E.; Kerr, W. L.; Yeh, Y. Nature 1987, 328, 241. (19) Wowk, B.; Leitl, E.; Rasch, C. M.; Mesbah-Karimi, N.; Harris, S. B.; Fahy, G. M. Cryobiology 2000, 40, 228.

CG025591E

Solution Interface Studied

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