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Mechanism for Helical Gel Formation from Evaporation of Colloidal Solutions Igor Veretennikov,*,† Alexandra Indeikina,‡ Hsueh-Chia Chang,‡ Manuel Marquez,§ Steven L. Suib,⊥ and Oscar Giraldo⊥ Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556; Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556; Los Alamos National Laboratory, Chemistry Division, Los Alamos, New Mexico 87545 and The Nanotechnology Laboratory, Kraft Foods R&D, 801 Waukegan Rd, Glenview, Illinois 60025; and U-60, Department of Chemistry, University of Connecticut, Storrs, Connecticut 06269-3060 Received May 24, 2002. In Final Form: August 30, 2002 We use MRI imaging to decipher the physical mechanism behind helical gel formation when a colloidal solution is evaporated from a small vertical or inclined capillary. A gel column, surrounded by the solvent, is observed to appear in the middle of a capillary. For nearly vertical capillaries, the denser gel column buckles under gravity to form a loose spiral. Further heating leads to the formation of a helical vapor pocket, surrounded by asymmetric liquid menisci. As the heating continues, this vapor pocket propagates downward and traces the buckled column. If gravity buckling occurs and if the maximum thickness of the annular solvent film is less than the solvent capillary length, significant nonuniform vapor pressure builds up within the vapor bubble because the vapor’s escape is obstructed and because the evaporation is nonuniform. This upward air pressure spiral is amplified by asymmetric menisci of the vapor pocket to produce a high liquid pressure gradient along the liquid spiral next to the helical vapor pocket. Both pressures are inversely proportional to the internal capillary diameter d, and together, they twist the buckled column to a much higher pitch. The balance of this force to the elastic force of the buckled column, which opposes coiling, leads to a minimum distance between pitches L that scales as d. When all the fluid outside the column has evaporated, the slow vapor release by the drying gel cannot provide sufficient pressure for coiling. Hence, the “compressed spring” starts to rewind and lengthen. The dominant force balance with the opposing dry friction force leads to a lower final pitch with an L ∼ d2 scaling. Both these scalings are consistent with our experimental data.
1. Introduction The self-assembly of colloidal particles into regular patterns after a colloidal solution evaporates has been a subject of intense interest recently. Stripes of colloidal array films are observed when a drop of colloidal solution evaporates on a substrate it wets.1 Colloids aggregate into wedding cake layers when they are driven by capillary forces of a convex thin film.2 Chemically patterned surfaces can also induce micron-level surface aggregation upon drying.3,4 Recently, a new helical colloid aggregation pattern is observed when a colloidal solution is evaporated from a small vertical open capillary.5 Such helices are only observed in capillaries with internal diameters smaller than 5 mm. In Figure 1, we show examples of helices produced by slow evaporation of colloidal solution of manganese oxide. As seen in Figure 1a, the number of turns per unit length depends on both the inner diameter of the capillary tube and the initial concentration of colloid. †
Department of Physics, University of Notre Dame. Department of Chemical Engineering, University of Notre Dame. § Los Alamos National Laboratory and Kraft Foods R&D. ⊥ University of Connecticut. * To whom correspondence should be addressed. ‡
(1) Adachi, E.; Dimitrov, A. S.; Nagayama, K. Langmuir 1995, 11, 1057-1060. (2) Xia Y.; Gates, B.; Yin, Y.; Lu, Y. Adv. Mater. 2000, 12, 693-713. (3) Aizenberg, J.; Braun, P. V.; Wilzius, P. Phys. Rev. Lett. 2000, 84, 2997-3000. (4) Ramos, L.; Lubensky, T. C.; Dan, N.; Nelson, P.; Weitz, D. A. Science 1999, 286, 2325-2328. (5) Giraldo, O.; Brock, S. L.; Marquez, M.; Suib, S. L.; Hillhouse, H.; Tsapatsis, M. Nature (London) 2000, 405, 38.
Figure 1. Helices produced by evaporation of colloidal solution of manganese oxide in vertical capillaries at T ) 80 °C. Concentrations of colloid C (M in Mn) are indicated. (a) Inner diameters d of large and small capillaries are 3 and 1.2 mm. (b) d ) 0.2 mm.
Even in microscopic capillaries shown in Figure 1b, we obtain very regular helices if the heating is sufficiently slow. The resulting helices do not depend on the kind of glass used as long as solvent wets the capillary. With MRI imaging, we investigate the mechanism behind this helical colloidal self-assembly phenomenon in this report. 2. Experimental Setup and Procedure Tetramethylammonium (TMA+) permanganate salts in 2-butanol/H2O are evaporated in open capillary tubes under constant heating from all sides at a controlled temperature T (typical
10.1021/la025987s CCC: $22.00 © 2002 American Chemical Society Published on Web 10/17/2002
Helical Gel Formation values are 80 and 85 °C). Sample holders allow us to heat up to 17 capillaries at different inclination angles (range from 20° to 90° in 10° increments) with respect to the horizontal. Two different NMR tubes with outer diameters o.d. of 5 and 2 mm are used in experiments with inclined capillaries (the corresponding inner diameters d are 4.25 and 0.9 mm). The MRI experiments have been carried out only with 5 mm NMR tubes at T ) 85 °C. The temperature inside the oven is measured by the glass thermometer. After a required waiting period, the sample is removed from the oven, sealed to prevent further evaporation, and cooled to room temperature. Images have been obtained by using Bruker AVANCE 300 MHz magnetic resonance imager. We use a standard three-dimensional spin-echo pulse sequence. The imaging parameters are echo time TE varies between 3 and 4 ms and repetition time TR varies from 750 ms up to 1000 ms. Four images are averaged to improve the signal-to-noise ratio due to a low signal intensity. The averaging significantly increases the acquisition time (3 h 25 min or 4 h 33 min for 256 × 64 × 64 data sets depending on the repetition time; 13 h 41 min for 256 × 128 × 128 data set up to 18 h 14 min for 512 × 128 × 128 data set). Spatial resolution varies from 47 up to 100 µm. After images have been acquired, the sample is returned to the oven. Control experiments do not show any difference between continuously heated samples and samples that have been used for imaging. Attempts to heat the sample inside the MRI magnet failed because nonuniform heating by the hot gas supplied from the bottom causes strong convection inside the capillary, which completely prevents colloidal particles from coagulation. The initial gel column formation requires about 15 min of heating inside the oven. In contrast, a few hours of heating inside the MRI magnet is not sufficient at the same temperature. The MRI signal decays with decreasing sample wetness because the imager acquires its signal from protons in the liquid phase. At the latest stages of drying, we use a high-resolution digital DALSA CA-D4 camera with 1024 × 1024 resolution. The camera is equipped with high-quality NIKKOR lenses and connected to a Dell Precision 620 workstation. The image analysis is carried out by using public domain NIH Image program (developed at the U.S. National Institute of Health) and Adobe Photoshop. For processing MRI images, we use ParaVision from Bruker BioSpin.
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Figure 2. “Eccentric rings”. Phase separation at early stage of evaporation. (a) Vertical slice through the capillary axis with arrows indicating location of cross sections in (b-d). The position of minimum thickness of the solvent film rotates around the column forming a loose spiral. Experimental parameters are C ) 0.1 M and t ) 1 h.
3. Experimental Results At our typical experimental conditions (T ) 85° C, o.d. ) 5 mm, concentration of colloidal solution C ) 0.1 M in Mn), phase separation begins after about 15 min of evaporation: a thin layer of clear solvent becomes visible at the top of the capillary, while a denser and richer colloid solution lies below. The thickness of the top solvent layer is about 5-10% of the initial height of the solution. The colloidal solution under the top meniscus below the top solvent layer is visually homogeneous. The MRI, however, discerns a distinct internal structure. In Figure 2, we present several slices of the MRI image of such solution after t ) 1 h of evaporation. A vertical slice through the capillary axis is shown in Figure 2a, and Figure 2b-d corresponds to the horizontal cross section at the indicated heights. It is evident that an internal column of gel, which is rich in colloid, has already formed at this stage, and it appears gray in the image. This internal gel column is surrounded by a thin annular film of the clear solvent (bright), which makes contact with (wets) the capillary. This thin film is invisible to the naked eye, but it creates a fluidlike appearance for the entire solution. It seems that the density difference between the gel and the solvent causes a gravitational buckling of the internal column: significant eccentricity between the circular gel cross section and the capillary is apparent at each cut, as seen in Figure 2b-d. The position of minimum thickness of the solvent film hence forms a loose spiral (about 1 pitch per 4.5 cm) on the capillary surface. With further heating, the internal column begins to shrink, both in length and in diameter. The volume fraction
Figure 3. Vapor pocket under the top meniscus. (a) Vertical slice through the capillary axis with arrows indicating location of cross sections in (c) and (d). (b) Vertical slice at 1.76 mm from the capillary axis, passing through the vapor pocket. (c) Horizontal slice across the vapor pocket with indicated location of vertical slices in (a) and (b). (d) Horizontal slice across the solvent stripe. Experimental parameters are C ) 0.1 M and t ) 6 h.
of the solvent relative to that of the condensed gel phase first increases slightly (since additional solvent is squeezed out of the gel) and then begins to decrease as the fluid evaporates. MRI images shown in Figure 3 correspond to 6 h of heating. The vertical slice in Figure 3a passes through the capillary axis, the slice in Figure 3b corresponds to one 1.76 mm from it, and cross sections in Figure 3c,d are made at two different heights. As seen in the figure, the buckled internal column is now slightly thinner
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Figure 4. “Rib cage” formed by solvent layers inside the colloidal solutions at initial stage of evaporation. Experimental parameters are C ) 0.05 M and t ) 1 h.
and, in several places, makes contact with the capillary. Its cross section remains circular and is of the same diameter along the entire imaging area. The decrease in width of the internal column in the upper part of Figure 3a is due to stronger twistingsthe thinner column simply exits the capillary axis (see also Figure 3c: the vertical slice in Figure 3a goes above the center of the buckled column). More profound buckling is also evident in Figure 3b: part of the out-of-center vertical slice passes only through the solvent. The dark area at the top right part of Figure 3b (and also at the top of the left bright solvent layer in Figure 3a) is a vapor pocket at the wide part of annular fluid film. Two liquid menisci, surrounding this vapor pocket, are clearly seen in Figure 3c. The air-liquid interface has a very complicated shape, roughly corresponding to two menisci which rotate around the column as parallel helices. At this stage, this helical vapor pocket is linked to the outside air above the top of the gel column, but it does not extend to the bottom of the capillary. As the heating continues, the vapor pocket propagates downward, following the buckled column and further twisting it. The bright “stripes” at the right part of the internal column in Figure 3a correspond to solvent layers inside the gel column. The horizontal slice in Figure 3d crosses one such layer. This phase separation inside the gel column is even more apparent in Figure 4, where we show vertical slice through the capillary axis (Figure 4a), horizontal slice (Figure 4b), and the maximum intensity projection of three-dimensional MRI image, viewed from two different angles (Figure 4c,d) for a less concentrated colloidal solution (C ) 0.05 M) at the early stage of evaporation. The origin of such “rib cage”, formed by layers of enhanced solvent concentration, is not clear to us, and its investigation is outside the scope of the present work. It seems that, for the more concentrated solution (C ) 0.1 M), the “rib cage” begins to form later (probably between 1 and 6 h and escapes our imaging) and disappears with further heatingsimages in Figure 5 do not show fluid layers inside the gel. After 8 h of heating, most of solvent has evaporateds there is no continuous fluid phase, while the three-
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Figure 5. Helical menisci. (a) Vertical slice through the capillary axis with arrows indicating location of cross sections in (b-d). The capillary wall and the curvature of the gel/air interface are shown in frame (c). The coiling is about twice stronger than the early stage in Figure 2 and is about one pitch per 2.5 cm. Experimental parameters are C ) 0.1 M and t ) 8 h.
dimensional coiling liquid menisci are clearly visible, at the places where the column touches the capillary wall. The cross sections in Figure 5c,d also show the gel column begins to lose its cylindrical symmetry: part of the interface that adheres to the capillary has the same curvature as the capillary wall, the radius of curvature of the gel-air interface is about 15% smaller, and the gel-solvent boundary provides a smooth connection between these two nearly constant curvature regions. Since the column continues to shrink and, at the same time, does not detach from the capillary wall, the number of pitches per unit length increases, as is evident from comparison of Figure 3 to Figure 5. Because drying at the bottom of the capillary seems to occur slower than at the top (probably, due to vapor transport limitations), the upper portion of helical meniscus evaporates first as heating continues. We observe that, at this stage, the helix typically attaches firmly to the capillary wall at the top, while its lower part can slide freely along the wall. As a result, the shrinking column slides up the wall of the capillary, such that the final solid helix does not touch the capillary bottom. Images in Figure 6 correspond to 15 h of heating, and this is the last stage of the helix formation process that can be imaged with MRI. After that, too little fluid remains inside the condensed phase and the MRI signal disappears. As seen in the figure, the fluid outside the column has completely evaporatedswe do not detect liquid menisci anywhere within the imaging area. It seems that both the increased stiffness of drying gel and the absence of lubricating fluid menisci around the contact points lead to the appearance of numerous internal cracks in the highstress regions (Figure 6a,b). The solvent accumulates within such defectssbright spots within cracks in Figure 6b,d correspond to higher fluid concentrations. The strongly asymmetric cross sections shown in Figure 6c,d indicate that the helix does not detach from the wall. The part attached to the capillary has about the same length as before (compare Figure 6c,d to Figure 5b-d), while the
Helical Gel Formation
Figure 6. Internal cracks. (a) Vertical slice through capillary axis. (b) Maximum intensity projection. (c) Horizontal slice with indicated position of the capillary wall and curvature of the gel/air interface. (d) Horizontal slice across the crack. The location of cross sections (c) and (d) is indicated in frame (a) by arrows. Experimental parameters are C ) 0.1 M and t ) 15 h.
Figure 7. Time evolution of the helix period L: (O) Data are for helices shown in Figures 2, 3, 5, and 6. Further heating leads to a lower final pitch. For C ) 0.1 M and o.d. ) 5 mm, the final value of L ) 60 mm is reached after about 90 h of heating. (b) Photograph of this dry helix shown in the last frame of Figure 9. Note the logarithm scale of the t-axis; L stays near it minimum value over a long duration.
radius of curvature of gel-air interface has decreased by about 25% (from 1.86 mm in Figure 5b-d to 1.34 mm in Figure 6c,d). Figure 7 illustrates how the coiling of the drying gel column changes during evaporation. Data on the distance L between pitches are obtained by tracing the positions of the gel columns using full MRI data sets because, in the best case, only about half of a period of each spiral is seen within an imaging area. Under further drying, the number of pitch per unit length (i.e., 1/L) first increases slightly and then begins to decreasesthere exists a specific degree of gel wetness, where coiling is at a maximum. For these experimental conditions (T ) 85 °C, o.d. ) 5 mm, and C
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Figure 8. Effect of capillary inclination angle on helix formation. Experimental parameters are C ) 0.075 M, t ) 96 h, and o.d. ) 5 mm. Inclination angles are indicated in the figure. Two frames for 20° inclination angle show the rectangular cross section of solid gel stripe.
) 0.1 M), the final pitch of the completely dry helices is about 4 times lower than that at maximum coilingsthe value of L increases to 60 mm after about 90 h of heating, and at t ) 96 h the solid gel stripe begins to crack. It should be emphasized that the time needed to reach the maximum coiling and/or final pitch, as well as the duration for the entire coiling process, depends not only on the above-mentioned experimental parameters but also on the initial depth of the fluid, the total length of the capillary, ventilation conditions inside the oven, etc. Hence, even identical temperature and time of heating do not ensure the same wetness of helices. While this makes comparison with earlier results5,6 difficult, the data shown in Figure 7 and our observations of subsequent drying process suggest that, near the maximum coiling, the helix period L change very slightly for a long time (about few tens of hours). The same seems to be true for dry gel stripessa few hours of nearly constant pitch exists before the cracking begins. This allows us to deduce some scalings for the minimum and final values of L during the entire coiling process, which will be discussed in the next section, and to use some of the photographs of helices presented in earlier reports5,6 to obtain more data on the helix period. To test the hypothesis that the primary coiling is due to gravity buckling, we make a few sets of experiments with capillaries installed at different inclination angles (range from 20° to 90° with 10° increments) with respect to the horizontal. The results for completely dry helices at two different concentrations of colloids are presented in Figures 8 and 9. As seen in the figures, at inclination angles below 40°, there is no coilingsthe initial column simply attaches to one side of the capillary and forms a solid gel stripe. At larger inclination angles, however, buckling is evident, and the coiling increases as the inclination angle approaches vertical. This confirms the importance of gravitational force in helix formation. Two frames presented in Figure 8a correspond to the same capillary, but rotated by 90°. The rectangular cross section (6) Marquez, M.; Robinson, J.; van Nostrand, V.; Schaefer, D.; Ryzhkov, L. R.; Lowe, W.; Suib, S. L. Chem. Mater. 2002, 14, 14931499.
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Figure 9. Effect of capillary inclination angle on helix formation. Experimental parameters are C ) 0.1 M, t ) 96 h, and o.d. ) 5 mm. Inclination angles are indicated in the figure.
of the final solid gel stripe is evident from its different widths on these two images. The situation is similar for all inclination anglesscompletely dry helices resemble “coiling ribbons”. We suggest that such final shape results from the higher flux of fluid from the side of the gel closer to the air interface and adhesion of the helix to the wall. Coiling is more profound for less concentrated solutions which is qualitatively consistent with earlier results5,6 and with helices obtained in smaller vertical capillaries shown in Figure 1. Quantitative comparison is, however, difficult, because for these experiments we use much shorter capillaries of relatively large diameters. This lower length/diameter ratio, which is up to 1 order of magnitude, affects both the heat/vapor transport in the capillary and the ability of initial column (which fills almost the entire tube, like that in Figure 2) to buckle under gravity. Also, the photographs in Figures 8 and 9 are made at significantly lower gel wetness than ones with different colloid concentrations in the earlier report5 and in Figure 1, and we have no opportunity to control this parameter. For smaller capillaries (with o.d. 2 mm) heated at T ) 85 °C, we do not find a well-defined dependence of the coiling on the capillary inclination angle. Moreover, the results are not reproducible for angles below 80°: sometimes, we obtain coiling at 20° and not at 70°. In some cases, especially at low inclination angles, the solid gel forms a plug at the top of capillary and prevents vapor from escaping. In such a case, the accumulated gas inside pushes the entire gel structure and all the liquid out of the capillary, thus breaking down all patterns formed. For vertical tubes and slower heating we, however, obtain very regular helices even in microscopic capillaries as ones shown in Figure 1b. We also did several experiments with square capillaries (with 5 mm side and C ) 0.1 M). We observe that the drying gel column always adheres to one side of the capillary. At an intermediate stage of heating, when the solvent outside the gel has already evaporated but the final dryness is not reached, the segment away from the capillary wall resembles a circular arc. Hence, vapor can always escape easily through the straight channels at the capillary corners, and we do not detect coiling in all these experiments. 4. Conclusions: Physical Mechanism for Helix Formation We propose that helical gel formation from evaporation of colloidal solution occurs in three stages dominated by
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different forces: (I) When the initial denser gel column is completely surrounded by the solvent, the coiling is due to gravity buckling. It creates a low-pitch helix, and the solvent spirals around it. The next stage cannot start without appreciable primary coiling at this stage. (II) Secondary twisting dominated by nonuniform vapor and capillary pressures in the surrounding asymmetric helical vapor pocket and liquid menisci. The result is a maximum in the pitch evolution. (III) Lengthening of the “compressed spring” as the gel dries, resulting in a lower final pitch. Below, we will discuss all these steps in detail and present a scaling theory for the second and third stage. We suggest that the primary coiling of the gel column (Figure 2) is due to gravity buckling, restricted by capillary walls. During this stage, the denser gel column is completely covered by the solvent, and gravity seems to be the only possible macroscopic driving force. This hypothesis is qualitatively supported by our experiments with inclined capillariesscoiling diminishes with lower gravitational pull (Figures 8 and 9). Further heating leads to the formation of helical vapor pocket surrounded by the asymmetric liquid menisci (Figure 3). Since this vapor pocket propagates from the top to the bottom of the capillary, we expect the radius of curvature for the fluid meniscus above the buckled gel column to be smaller than that of the lower meniscus. This leads to the downward increase in capillary pressure within the fluid phase. While limited field of view and insufficient spatial resolution of our MRI images do not allow us to make reliable measurements of curvatures of air/liquid interfaces at different positions, the asymmetric shape of fluid meniscus is apparent in Figure 3b, and even a slight difference in length and curvature of the bright fluid segments may be seen in Figures 3c and 5. Moreover, while the growing helical vapor pocket is linked to above atmosphere, the escape of gas released by internal evaporation is obstructed by the buckled gel column. This leads to an increase in the vapor pressure at the lower part of the vapor pocket. Both liquid capillary and gas pressures are transmitted through the helical internal menisci as a torsion force that twists the original loose spiral into a higher pitch along the same plane. This secondary twisting, however, cannot start without appreciable gravity buckling. Indeed, if the primary coiling is insufficient (low inclination angle, large diameter-tolength ratio for wide capillaries, or shape-induced high rigidity of initial gel column in square capillaries), the asymmetry of fluid menisci (and hence the capillary pressure gradient) diminishes. Also, the resulting straight channel will allow the vapor to escape without building up a pressure to twist the gel column. Thus, all driving forces for secondary coiling would be removed. The elevated air pressure within the vapor pocket may also have significant variation along the pocket due to nonuniform evaporation rates. Evaporation should favor the wide side of annular film with higher solvent fraction and wider gaps to avoid capillary suppression of evaporation. However, if the maximum thickness of the annular film near the tip of propagating vapor pocket is larger than the capillary length H ) (σ/Fg)1/2, which is about 2.5 mm, the asymmetry of fluid menisci again vanishessall radii of curvatures become of the order of H regardless of their locations. While this will eliminate one of the driving forces for secondary twisting, the condition that the gap width is smaller than H seems to be a weaker restriction than the necessity of appreciable gravity buckling. The latter, however, cannot be estimated without knowing the mechanical properties of the gel column.
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When most of solvent is evaporated (Figure 5) and the gel column becomes firmly attached to the capillary wall at the top, the helical liquid menisci does not allow the column to detach from the wall, but the column is permitted to slide along the capillary. The process is qualitatively similar to a wet rope coiling along walls of a bucket under the action of gravity except that the driving forces are now the difference in capillary pressures for two asymmetric helical menisci and the pressure of exiting vapor and both acts upward. (Both pressures are inversely proportional to the instantaneous column diameter d* and hence the internal capillary diameter d, because of ideal gas law for the gas pressure and because of the curvature considerations for the liquid capillary pressure.) Hence, the coiling continues until the fluid outside the column completely evaporates. To estimate the minimum period of the spiral L, we should balance this net driving force to the elastic torsion force of the column, which opposes coiling. The driving force can be estimated as a product of the net pressure and half of the surface area of the helix,
Fd ∼
1 (d l) ∼ l d* *
Figure 10. Effect of concentration of colloidal solution C on helix period L. Data are taken using photographs presented in ref 5.
(1)
where the length of one pitch
l ) (L2 + [π(d - d*)]2)1/2 ≈ L
(2)
is close to the distance L between the pitches for our typical low-pitch helices. Considering the coiling gel column as a compressed spring, we obtain the scaling for the opposing elastic force,
Fe ∼
C*d*2 Cd2 ≈ l l
(3)
where C* is the instant concentration of colloid inside the gel. We also assume that the shrinking occurs mostly in the radial direction, such that C*d*2 ≈ Cd2 due to mass conservation for the colloid. For low-inertia twisting, Fd ≈ Fe and, from eqs 1-3, one obtains the scaling for the helix period L,
L∼d
(4)
at the gel wetness corresponding to maximum coiling. For low-pitch helices l ≈ L, and scalings (1) and (3) also suggest L ∼ C1/2 for the dominant dependence of L on the initial colloid concentration. Of course, the entire L-C dependence may not be captured by above simple scaling arguments because vapor and capillary pressures, in general, depend on C* (and hence on initial concentration C). It is possible that the actual dependence of elastic force on concentration is more complicated than eq 3. However, analysis of earlier experimental data of Giraldo et al. (taken at the same heating time t ) 24 h, temperature T ) 85 °C, internal capillary diameter d ) 1.2 mm, and varying colloid concentrations) confirms this general scaling. As seen in Figure 10, the dependence of helix period on concentration are well represented by a straight line,
L-1 ) k1C-1/2 - k2
(5)
despite the fact that all helices in this data set correspond to lower gel wetness than that at the maximum coiling (i.e., these data points belong to the inclined portions of
Figure 11. Effect of internal capillary diameter d on helix period L. Scaling is linear when coiling is at the maximum, near the minimum L range in Figure 7 (open symbols). The periods of completely dry helices (solid symbols, including the last data point of Figure 7) in relatively large capillaries scale quadratically with respect to d. (0, 9) Reference 5. (4) Reference 6. (O, b) Present work.
the appropriate L-t curves, analogous to that shown in Figure 7). This, however, is not very surprising, because the physical mechanism for secondary twisting remains the same as long as coiling continues. While the coefficients k1 ) 0.092 ( 0.004 mm-1 M1/2 and k2 ) 0.239 ( 0.002 mm-1, determined from fitting the data in Figure 10, depend on capillary diameter, heating time, and possibly on other experimental parameters, combining scaling (4) with empirical dependence (5) suggests that
L)k
dC1/2 C ˜ - C1/2 1/2
(6)
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The effective concentration C ˜ and the nondimensional proportionality coefficient k are independent of d, and their dependence on time etc. should vanish at maximum coiling when the variation is slow. At the experimental conditions of Figure 10, we find C ˜ ) (k1/k2)2 ) 0.148 ( 0.006 M, and at low colloid concentrations, we recover the simple scaling L ∼ dC1/2. At later stages, when all fluid outside the column is evaporated and the slow vapor release by the drying gel cannot provide sufficient pressure for coiling, the “compressed spring” may start to lengthen. In the absence of fluid meniscus, the opposing dry friction force should remain nearly constant and, for sufficiently large capillaries, is independent of helix dimensions. Hence, the scaling for the final pitch follows from the condition Fe ≈ constant, and from eqs 2 and 3 we obtain
L ∼ d2
(7)
for low-pitch completely dry helices. Figure 11 shows the dependence of measured helix period L on the internal capillary diameter d. Data are taken using the enlarged version of photographs presented in earlier works,5,6 our photographs of completely dry helices (frames corresponding to 90° inclination angle in Figures 8 and 9) and full MRI data set at t ) 15 h (sample slices are shown in Figure 6). All these data, except one in Figure 8, are taken at the same initial colloid concentration C ) 0.1 M. To scale this point (taken at concentration C0 ) 0.075 M), we use a similarity variable, suggested by eq 6. Because C ˜ corresponds to the maximum initial concentration that allows coiling, we expect that it depends on time and other varying experimental parameters more weakly than k or may even be time-
independent. Hence, we use the effective capillary diameter
( )
d ˜ )d
C0 C
1/2
C ˜ 1/2 - C1/2 C ˜ 1/2 - C01/2
(8)
which is equal to 2.3 mm instead of the actual 4.25 mm, to put this point on the same plot. As seen in the figure, data for helices at wetnesses which roughly correspond to maximum coiling (open symbols) and data for completely dry helices (solid symbols) clearly form two different lines. The scaled data point (middle black circle) is consistent with other data for dry helices, which support our assumption about the time/parameters independence of the effective concentration C ˜ . Two solid lines, drawn through the data points, are obtained by the mean-square fitting based on scalings (4) and (7). The agreement with our scaling theory is quite good, despite the fact that we cannot control the wetness of helices directly. This provides another support to the suggested physical mechanism for helical gel formation. We expect this helical colloid aggregation phenomenon to be generic to evaporation in small (diameter-to-length ratio should allow primary gravity buckling) vertical capillaries heated from the side. Acknowledgment. The financial support of Kraft Foods is greatly appreciated. I.V. also thanks Prof. James Glazier for providing almost unlimited access to MRI imager. MRI facility of University of Notre Dame have been partially supported by DOE Grant DE-FG02-99ER45785 and NASA Grant NAG3-2366. LA025987S
