On the Creeping of Saturated Salt Solutions - Crystal Growth & Design

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The creeping of Saturated Salt Solutions Elucidated Willem J. P. van Enckevort, and Jan H. Los Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/cg301429g • Publication Date (Web): 04 Apr 2013 Downloaded from http://pubs.acs.org on April 8, 2013

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Crystal Growth & Design

On the Creeping of Saturated Salt Solutions Willem J.P. van Enckevorta* and Jan H. Losa,b a

b

Radboud University Nijmegen, Institute for Molecules and Materials Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Institute of Physical Chemistry and Center of Computational Sciences Johannes Gutenberg University Mainz Staudinger Weg 9, D-55128 Mainz, Germany

Abstract Creeping is a well known, but annoying phenomenon in the preparation of crystals from solution, where growing crystallites gradually extend up the walls of the growth vessel. In this process solution is transported towards the tip of the creeping crystallites, where solvent evaporation takes place and solid material is deposited. In this study the growth of crystal aggregates extending from evaporating droplets of saturated aqueous solutions of ionic salts, placed on different substrate materials has been investigated using optical microscopy. It is shown that the rate determining step of the crystallization process is the evaporation of solution, following Fick’s laws. Fresh solution, necessary to continue the growth process is supplied by liquid flow along the growing crystallites. This can take place aside and on top of the crystallites (top supplied creeping, TSC) or in the narrow space between the crystallites and the substrate (bottom supplied creeping, BSC). The occurrence, mode (TSC or BSC) and velocity of creeping is shown to be determined by the relative humidity of the ambient air, the various interfacial energies involved and the shape and size of the growing crystallites. In a number of cases seaweed-like patterns are formed by repeated side-branching of the growing aggregates, induced by 3D nucleation of secondary crystallites.

*

Corresponding author; e-mail address: [email protected], tel: +31-24-3653433, fax: +31-243653067.

-1ACS Paragon Plus Environment


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On the Creeping of Saturated Salt Solutions

Willem J.P. van Enckevorta* and Jan H. Losa,b

a

Radboud University Nijmegen, Institute for Molecules and Materials Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

b

Institute of Physical Chemistry and Center of Computational Sciences Johannes Gutenberg University Mainz Staudinger Weg 9, D-55128 Mainz, Germany

Abstract Creeping is a well known, but annoying phenomenon in the preparation of crystals from solution, where growing crystallites gradually extend up the walls of the growth vessel. In this process solution is transported towards the tip of the creeping crystallites, where solvent evaporation takes place and solid material is deposited. In this study the growth of crystal aggregates extending from evaporating droplets of saturated aqueous solutions of ionic salts, placed on different substrate materials has been investigated using optical microscopy. It is shown that the rate determining step of the crystallization process is the evaporation of solution, following Fick’s laws. Fresh solution, necessary to continue the growth process is supplied by liquid flow along the growing crystallites. This can take place aside and on top of the crystallites (top supplied creeping, TSC) or in the narrow space between the crystallites and the substrate (bottom supplied creeping, BSC). The occurrence, mode (TSC or BSC) and velocity of creeping is shown to be determined by the relative humidity of the ambient air, the various interfacial energies involved and the shape and size of the growing crystallites. In a number of cases seaweed-like patterns are formed by repeated side-branching of the growing aggregates, induced by 3D nucleation of secondary crystallites.

*

Corresponding author; e-mail address: [email protected], tel: +31-24-3653433, fax: +31-243653067.


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1. Introduction

Creeping is a well known, but annoying phenomenon in the preparation of crystals from solutions, where a cake or crust of solid material gradually extends up the glass walls of the growth vessel [1] (figure 1). In this process the solution is transported towards the top of the creeping solids, where solvent evaporation takes place and solute material is deposited. In the use of evaporative crystallization techniques, both on laboratory and industrial scale, the formation of such a crust layer on the walls and the stirrer shaft of the crystallizer may impose major problems [2]. Despite its relevance, apart from a few exceptions [3-5], creeping of saturated solutions received little attention in the scientific literature during the last decades. Most studies on this issue were published in the first part of the previous century [1,6-9]. Washborn [6] suggested that solution transport is induced by capillary action, in which solution is drawn upwards to the growth front through the narrow spaces between adjacent crystallites or between the crystallites and the glass wall. However, Hazlehurst, Martin and Brewer [9] pointed out that the growing crystals are in close contact with the glass wall and that the solution must be transported over, instead of under the crystallites. Creeping was found to occur in varying extent for many substances, like NH4Cl, ZnSO4, BaCl2, CuSO4, and K2Cr2O7 in saturated aqueous solutions as well as for sulfur dissolved in CS2. Creeping is the evaporation driven extension of crystals on solid, non-porous substrates. Related phenomena are subflorescence and efflorescence in and on top of porous materials, which have been studied extensively in view of the conservation of (historical) monuments and buildings made of porous stone materials [4, 10-20]. Subflorescence refers to the growth of crystals in pores, which as a consequence of crystallization pressure leads to degradation of stone material [14,15,16,18,20]. Efflorescence is the 3D growth of crystal aggregates on top of the porous materials and is driven by evaporation of the water molecules from



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the salt solution emerging at the surface [4,12,13]. Efflorescence reduces the degradation of the stone material as solution is transported to its surface preventing harmful crystallization in its porous inner part [4,12,13]. Crystallization processes on non-porous and porous substrates governed by solvent evaporation have been compared in a 1989 study by Zehnder and Arnold [11]. As will be exemplified in our study, the rate determining step in many creeping processes is the evaporation of solvent at the periphery of the polycrystalline aggregates. In contrast to creeping, mass transport limited growth of crystals from solution received considerable attention in the literature [21-23]. In this case solute transport is governed by Fick’s second law

∂c = D∇ 2 c , ∂t

(1)

Figure 1. Creeping from a saturated aqueous NaCl solution (with Fe(CN)64- additive in the right-hand figure), resulting in an extended crust of crystals extending up the glass walls of


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the beaker. Left: Primary (“mural”) creeping; Right: Secondary (efflorescent) creeping following primary creeping.

with c the solute concentration and D the solute diffusion constant. The development of the growing crystal can be described by solving this differential equation with the appropriate moving- boundary conditions, which are determined by the time dependent position of the growing crystal surface, its surface energy and its surface kinetics [21,22]. It has been shown that if the crystal size exceeds a certain critical value, then morphological instability sets in and branched growth forms develop [23]. Using the nomenclature introduced by Brener et al. [24 - 28] branched crystal patterns can be classified into four morphologies, namely compact dendrites, fractal dendrites, compact seaweed (also known as “dense branched morphology” (DBM) [27]) and fractal seaweed. The term “seaweed” refers to patterns that do not show a pronounced orientational order, i.e. branching occurs in random, non-crystallographic directions. This is in contrast to “dendritic” patterns, which show branching in preferred crystallographic directions. “Fractal” refers to a non-integer Hausdorff dimension of the growth pattern; compact (termed “homogeneous in ref. [29]) indicates an integer Hausdorff dimension [24 - 29]. Highly ramified patterns preferentially develop at high supersaturations and low surface energies. During the last few decades a large number of different growth patterns have been found and investigated, both by computer modeling [29,30] and by experiment [21, 31-33]. During experiments in our laboratory we found that upon evaporating droplets of saturated aqueous K2Cr2O7 solution from a glass surface, extended branched seaweed patterns are formed by creeping. As shown in figure 2a these features radiate from the edge of the droplet and expand over the glass surface. A similar branched pattern, but now emerging from an evaporating droplet of saturated NaCl solution with K4Fe(CN)6 additive, was recent-



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ly observed by Gupta et al. [13]. In the present study we investigate this and other phenomena of pattern formation by creeping using in-situ and ex-situ optical microscopy of evaporating droplets of saturated salt solutions. In contrast to the work by Takhistov and Chang [34], who examined stain morphologies left by evaporating droplets, we here concentrate on growth patterns protruding outwards from the droplet edges. Several model systems are explored. The patterns created are described and interpreted in terms of crystal nucleation and growth, solute flow, solvent evaporation and morphological instability. In this text only primary (or “mural” [9]; Figure 1a) creeping, i.e. the direct growth of the crystals on the substrate surface, will be treated. No attention will be given to secondary (or “efflorescent” [9]; Figure 1b) creeping, which is creeping upon previously deposited crystals.

2. Experimental

Droplets of saturated aqueous solutions of several inorganic salts, such a K2Cr2O7, (NH4)2Cr2O7, CdI2, KH2PO4, (NH4)H2PO4, KCl, CuSO4.5H2O, ZnSO4, and K-alum are placed on carefully cleaned microscope glass slides or other substrate materials. The substrates are cleaned by rinsing with deionized water followed by rubbing with ethanol using a paper tissue to remove organic contamination. The contact angle of water drops on the glass substrates prepared in this way is 15 to 20°. Typical droplet diameters used in the experiments range from 6 to 10 mm. The process of droplet evaporation and the associated creeping are observed in-situ by reflection (mostly differential interference contrast or dark field) optical microscopy. The images are recorded by video; high resolution still images are obtained by a CCD camera. After complete evaporation, an overview of the creeping patterns is obtained by low magnification optical transmission microscopy. The experiments are carried


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Figure 2. Seaweed patterns of K2Cr2O7 crystal aggregates formed upon evaporation of a droplet of (super)saturated aqueous K2Cr2O7 solution from a glass substrate: (a) low magnification view of a typical pattern; (b) higher magnification view of a smaller aggregate showing individual crystallites.

out at ambient conditions, which means in air (relative humidity: 25 – 50%) and at room temperature. Only pro-analysi chemicals are used.



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3. Observations

3.1. K2Cr2O7 solutions The most well developed branched aggregate patterns were obtained by evaporation of droplets of (super)saturated aqueous K2Cr2O7 solution on a glass substrate. A typical example is given in figure 2a. Figure 2b shows an ex-situ micrograph of a smaller creeping pattern at a higher magnification. It can be seen clearly that this aggregate is composed of numerous individual K2Cr2O7 crystallites. The aggregate trees originate from one single crystallite protruding from the rim of the droplet. Repeated nucleation of new crystallites and their subsequent growth lead to the characteristic branching patterns. Apart from a very weak crystallographic alignment along the directions of the aggregate arms, transmission polarization microscopy showed no preferred crystallite orientation. Furthermore, the pattern as a whole does not show any orientational order, which implies that it must be classified as a skeletonseaweed structure. In-situ observation of the growing patterns at highest magnifications provides a more detailed insight in the mechanism of K2Cr2O7 creeping. Figure 3 shows that the growing aggregate is embedded by a layer of solution, which follows the contours of the growing crystallites because of surface tension effects. Near the rapidly growing tips the solution evaporation is maximal and its layer thickness is minimal. From the observations it is clear that maximal growth occurs at the tips of the aggregates and that fresh solution is transported from the droplet edge towards the tip along the liquid layer embedding the aggregates. This process is schematized in figure 4. The growth rate at the periphery of the aggregates is typically ∼2 µm/s. The in-situ observations also showed that creeping growth of thin needles and thin platelets proceeds faster than that of thick, block-shaped crystals.


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Figure 3. In-situ observation at high magnification of a growing K2Cr2O7 seaweed pattern embedded by a thin layer of solution.

Figure 4. Schematic representation of K2Cr2O7 crystal aggregate formation by creeping. -9ACS Paragon Plus Environment


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The roughness of the substrate is not a key factor for the occurrence or absence of creeping. Droplets of evaporating K2Cr2O7 solution on polished glass plates as well as on frosted glass produce seaweed patterns by creeping. However, for the frosted glass surfaces the patterns are more compact and are composed of smaller, thin crystallites. Careful in situ examination of the growing tips at highest magnifications showed that here crystal growth is largely determined by solution transport between the crystallites and the substrate. This follows from the direct observation of liquid underneath the crystallites as well as of steps moving on the lower surface just behind the tip (figure 5).

Figure 5. In-situ observation of a growing K2Cr2O7 crystallite by bottom supplied creeping.

For growth on smooth glass substrates solution transport over and along the crystallites as well as underneath was observed. In the following we shall denote the first case as “top supplied creeping” (TSC) and if transport takes place between the crystallites and the substrate as “bottom supplied creeping” (BSC). It is interesting to note that both possibilities have been advocated in the older literature: Washborn [6] favoring BSC and Hazlehurst [9] TSC.


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Comparison of the various patterns shows that if TSC is the dominant mechanism of growth, the aggregates show a more open structure with wider spaced branches; if creeping is determined by BSC the aggregates are more compact.

Figure 6. In-situ micrographs showing the development of K2Cr2O7 creeping patterns. The time interval between the recording of the successive images is 180 seconds.



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Figure 6 shows the time dependent development of the K2Cr2O7 creeping patterns. As already mentioned by Hazlehurst et al. [9] the initial crystallization starts in the droplet close to its rim, bulges the edge of the droplet and then the aggregate grows outwards by creeping. This indicates that at least the first few crystallites must adhere to the glass plate, otherwise they would be drawn backwards into the droplet and no creeping occurs [9]. Indeed, in a few cases, especially for KCl solutions, we found crystals freely floating in the solution which tried to penetrate the droplet rim by growth outwards, but these were pulled back in the liquid by capillary forces. The majority of the K2Cr2O7 crystallites forming the aggregate were not fixed to the substrate during growth: by gently pushing with a thin needle they were easily displaced, floating on a thin layer of water. A few crystallites adhered to the glass substrate, presumably these kept the others at a fixed position during growth. To determine the rate-limiting step of the creeping process a gentle flow of dry nitrogen was applied to the evaporating droplets of K2Cr2O7 solution. This increased the velocity of creeping several times and demonstrates that the rate of aggregate formation is limited by the mass transport of evaporated water molecules in air. This was confirmed by the observation that very slow evaporation of a droplet of (super)saturated aqueous K2Cr2O7 solution in a nearly closed vessel hardly lead to creeping. Only larger crystallites within the droplet were formed. The evaporation driven crystal growth described here is analogous to the efflorescence on top of porous materials, where volume diffusion of evaporating water molecules determines crystal growth and in which fresh solution is supplied via the network of narrow spaces in the growing 3D crystal aggregates (4, 12, 13). Another important factor that affects creeping is the wetting of the solution with the crystal and the substrate surface. Evaporation of a K2Cr2O7 solution droplet on a smooth Teflon surface showed the development of numerous crystallites in the droplet, but no creeping. It was found that due to the poor wetting with this substrate, no water layer develops at the


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periphery of the crystallites that attempt to protrude the droplet at its rim. The amount of wetting was easily verified by estimating the contact angle of the solution droplet on the substrate: for glass it was about 25° and for Teflon about 75°. In attempting to introduce some orientational ordering in the creeping patterns, experiments were carried out using single crystalline substrates, such as {0001} quartz, freshly cleaved mica plates and {100} calcite. No ordering was found in any case and the patterns remained seaweed-type instead of becoming dendritic.

(NH4)2Cr2O7 shows a similar creeping behaviour as K2Cr2O7, except that the crystals grow as needles, by TSC, which leads to spherulite-like growth as shown in figure 7. After growth, presumably due to Ostwald ripening [35] or maybe due to a polymorphic phase transition, recrystallization occurred, which resulted in aggregates composed of block shaped crystallites with morphologies characteristic of (NH4)2Cr2O7.

Figure 7. Spherulite-like growth of (NH4)2Cr2O7 enhanced by top supplied creeping. At the left, the needles are disintegrated by recrystallization effects.



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3.2. KH2PO4 solutions Evaporation of a droplet containing a (super)saturated aqueous solution of KH2PO4 on a glass substrate gives extensive creeping, which leads to an open structured pattern. As shown in figure 8a, the pattern is composed of rods (lower right), ≈ 100 µm wide, and very thin needles (left) less than 20 µm thick. The rods exhibit a morphology which is typical for the stable, tetragonal form of KH2PO4 [36]. The length direction is and the crystals are bounded by {100} and {101} faces. However, the shape of the thin needles is not consistent with the tetragonal KH2PO4 structure. Moreover, if a growing rod approaches a group of thin needles the second type readily dissolves. This phenomenon cannot be explained by Ostwald ripening, because a needle was never found to develop into a rod and extremely small rods

Figure 8. (a) Creeping pattern of KH2PO4, composed of two different polymorphs, formed upon evaporation of a droplet of saturated solution from a glass substrate. (b) In-situ image of a growing KH2PO4 needle embedded by a thin layer of solution.

were found to grow in the presence of dissolving needle crystals. Presumably, the thin needle crystals are a metastable (pseudo)polymorph of potassium dihydrogen phosphate, which


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needs further identification. Attempts to isolate individual needles for X-ray diffraction analysis failed, because the crystals are highly unstable and readily convert into the stable form of KH2PO4.

The growth of both types of KH2PO4 crystals clearly proceeds by the TSC mechanism. As can be seen in figure 8b a liquid layer of ∼50 µm thickness embeds a thinner needle of the stable polymorph, which protrudes from the drop edge. Application of a flow of dry nitrogen dramatically increases the creeping rate from ≈ 2 to ≈ 10 µm/s. The growth rate also increases strongly for decreasing needle thickness. As will be elaborated further on in this paper, both observations indicate that the pattern growth is limited by diffusion of the evaporating solvent. For (NH4)H2PO4, only very little creeping was observed. The few crystals protruding the rim of the droplet were thick, block shaped and grew slowly. Here no metastable polymorph was found.

3.3. Other salts 3.3.1. CdI2: anti-creeping Upon evaporation of a droplet of (super)saturated CdI2 solution on a glass plate, no creeping was encountered. In fact the opposite happened, namely platy, dendritic crystallites that were nucleated at the rim of the droplet grew inwards, and thereby “pushing” the evaporating droplet inwards as well (figure 9a). This phenomenon of centering instead of expanding the solution by crystal growth will be termed “anti-creeping” in the following. Removal of grown crystallites near the rim of the droplet did not reveal any solution layer between the crystals and the glass substrate. The crystallites adhered well to the substrate. Moreover, no liquid was found on top or in front of the grown crystallites. As will be elaborated in the dis-



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cussion part of this paper, anti-creeping is induced by poor wetting of the saturated solution with the growing crystal surfaces. This poor wetting is demonstrated by the repulsion of the solution at the rim of the droplet by growing crystallites, as shown in figure 9b.

Figure 9. Anti-creeping of CdI2: (a) schematic representation; (b) in-situ image of a CdI2 crystallite repelling the liquid at the rim of the solution droplet.

3.3.2. BaCl2.2H2O and ZnSO4 Evaporation of droplets of saturated, aqueous BaCl2 solutions results in very little creeping; for ZnSO4 no creeping is found at all. In both cases numerous crystallites are formed near the rim of the evaporating droplet, “building” a close, solid wall following the shape of the droplet surface. This wall blocks transport of liquid to the periphery and thus prevents creeping. This implies that the wall must be in close contact with the glass plate and no liquid layer occurs on its surface, thus preventing BSC and TSC respectively.


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3.3.3. NH4Cl Evaporation of an aqueous NH4Cl droplet produced creeping patterns that are typical for BSC, followed by secondary creeping leading to thickness growth. The patterns formed are very dense, approaching complete surface coverage (figure 10). The aggregates are very fine grained, with crystallite sizes of maximally 10 µm. Careful examination by high magnification in-situ optical microscopy viewing from below through the glass substrate showed the presence of a thin water layer between the crystallites and the substrate. Moreover, a soft touch displaces the aggregate very easily over the substrate. This demonstrates that BSC is the dominant mechanism of primary creeping. The growth fronts are typically 0.5 µm thick as could be judged from interference patterns. The creeping rate increases for thinner growth fronts. Behind the growth front, growth solution penetrates between and moves over the crystallites, leading to thickness growth by secondary creeping.

Figure 10 Bottom supplied creeping growth of NH4Cl giving dense, compact patterns.



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3.3.4. KCl, KAl(SO4)2.12H2O and CuSO4.5H2O These salts show a tendency to creeping, but hardly any seaweed patterns are formed. In fact, potash alum only shows creeping after addition of a small amount of soap as a detergent to the solution, which lowers its surface tension. KCl and especially potash alum form thick, block shaped crystallites, which slowly protrude from the droplet rim. Often the crystallites are pulled back into the droplet by capillary forces. The crystallites are embedded by a thick layer of liquid (figure 11), which indicates that growth is determined by TSC. The growth rate of these thick crystallites is low (∼0.5 µm/sec). For potash alum no and for KCl very little secondary nucleation is encountered, which is of essential importance for the development of branched seaweed patterns. The shape of the crystallites is block-like (in fact: cubo-octahedral for alum and rectangular-cubic for KCl), which

Figure 11. Top supplied creeping growth of a KCl crystal. Due to its large thickness, the growth is slow. No side-branches develop, because of a lack of secondary 3D nucleation.

means that growth takes place far below the critical supersaturation for kinetic roughening [37]. This implies that interface kinetics remains important. It also points to a relatively high


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crystal-fluid surface energy, which again explains the difficulty of forming the threedimensional nuclei needed for side-branching. Copper sulfate behaves in a similar way as KCl and potash alum. However, instead of blocks now thick rods are formed, which stick out from the droplet rim. The growth rate is not high (∼1 µm/sec) and the rods are embedded by a solution layer, which again indicates TSC. Since no secondary nuclei are formed, no seaweed pattern develops.

The observations for the various salts are summarized in table 1.

Table 1: Creeping properties of the various salts investigated

Salt

Creeping property

Aggregate shape

K2Cr2O7

TSC and some BSC

seaweed

NH4Cr2O7

TSC

spherulite-like, followed by recrystallization

KH2PO4

TSC

Two polymorphs: i) rods: skeleton-seaweed ii) thin needles: spherulite-like

CdI2

anti-creeping

replaces droplet area

BaCl2, ZnSO4

no creeping

overgrows droplet surface

KCl, KAl(SO4)2.12H2O,

TSC

block shaped single crystals; no

CuSO4.5H2O, NH4H2PO4

branching



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4. Mechanisms of creeping

From the experiments it is clear that creeping is a complex phenomenon with many possible variations. The crystal growth process depends on many parameters, such as water evaporation, the various surface energies involved, the occurrence and rate of secondary nucleation, crystal size and shape and others. Therefore, a complete quantitative description of the whole process is practically impossible. Nevertheless, several important aspects will be elaborated below. Only primary creeping will be considered.

4.1. Top supplied creeping 4.1.1. Growth rate In the case of KH2PO4 and to a large extent K2Cr2O7 and (NH4)2Cr2O7, the solution necessary for creeping growth is transported along and over the crystal surfaces (figure 12a). The situation is reminiscent of the case of a spilled coffee drop drying on a solid surface, resulting in the transport of dispersed solids to the rim of the droplet [38-41]. However, in our case only solution is transported, which evaporates at the edge of the droplet, leading to laterally expanding creeping patterns by crystal growth. It is shown that the creeping rate is determined by water vapor transport in air, which follows Fick’s second law, given by equation (1). Taking the stationary case,

∂c = 0 , one obtains ∂t

∇ 2c = 0

(2a)

or, approximating the liquid envelopment around the tip as a hemisphere, in spherical coordinates [42]


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d 2 c 2 dc + = 0, dr 2 r dr

(2b)

with c(r) the water vapor concentration in air and r the radial coordinate perpendicular to the

Figure 12. Transport of solution and subsequent evaporation of solvent near the growing crystal tip during: (a) top supplied creeping; (b) bottom supplied creeping. In (a) the liquid layer at the side faces of the needle is not shown.

liquid tip surface. Solving equation (2b) for a tip radius Rt (figure 12a) leads to [42]

c( r ) =

Rt (cs − cb ) + cb , r

(3)



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with cb the water vapor concentration in air at r→∞, i.e. the ambient vapor pressure, and cs the water vapor concentration near the liquid surface, which is close to its equilibrium value (cs ≅ ceq). Using Fick’s first law, J = − D

dc , and integrating over the hemisphere gives a todr

tal water evaporation rate of

J tot ≅ CDRt (cs − cb ) ,

(4)

with C a geometrical factor equal to 2π. If one considers the influence of the evaporating solvent behind the tip, C ≈ π-2π. For a non-hemispherical tip with a solid-liquid contact angle θ ≠ 90°, the constant C is a function of θ as derived by Picknett and Bexon [43]. Taking the

cross section of the growing crystal tip as A, one obtains a tip propagation rate of

vtip =

J tot s , A ρ

(5)

with s the solubility in kg per kg solvent and ρ the specific mass of the crystal. If one assumes the liquid layer having a thickness of ε, (or Rt ≅

vtip ≅

A + ε ) one finally obtains

CDs( A + ε )(cs − cb ) . Aρ

(6)

This equation confirms that the growth velocity increases for decreasing tip radius,

A , and

that dry nitrogen, i.e. cb → 0, enhances the creeping rate. It is clear that the driving force for the whole creeping process is given by the difference of the equilibrium pressure of water at


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the temperature of the liquid droplet and the actual vapor pressure in the ambient air: ceq - cb. In deriving equation (6) it is assumed that the ambient air is stagnant; the occurrence of gas flow introduces a finite boundary layer, δ, which increases vtip. Another factor that might influence the evaporation rate of the solvent near a creeping needle tip is an increased vapor pressure introduced by the curved surface of the liquid. The dependence of vapor pressure, P(Rt), on liquid curvature, Rt, is given by Kelvin’s law [44]:

ln

P( Rt ) 1 2Ωσ = , Peq kT Rt

(7)

with Peq = P(Rt→∞), σ the surface energy of the liquid and Ω the volume of one liquid molecule. From this equation it follows that the partial vapor concentration cs(Rt) ∝ P(Rt) increases for decreasing Rt. Using equation (6) this again implies that vtip increases for smaller tip radii. At 25 °C, σ = 72 mJ/m2 for water. From the observed tip radii of one to a few tens of microns, it follows that the relative increase of vapor concentration at the tip surfaces, (csceq)/ceq, amounts 10-3 to 10-2. Since this small increase is far too low to account for the enhanced growth of protruding needle tips, it can be concluded that creeping is determined by mass transport and that the Kelvin effect plays a negligible role.

4.1.2. Wetting condition The condition for the occurrence of TSC is largely governed by the various surface energies involved. If the occurrence of a liquid layer stretching along the edge between crystal and substrate is energetically favorable, then solution will be transported from the central droplet, along the edges of the aggregate crystals, towards the aggregate tips. In this case,



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TSC will take place. If not, then the liquid is repelled backward to the central droplet and creeping will not occur. In deriving the energy condition for the presence of this liquid layer explicitly, it is assumed that the liquid is stretched along the crystal-substrate edge over an infinite distance. This makes the problem two-dimensional and reduces Laplace’s equation [45] to ∆ P = γ lg (1 / r ) . Here ∆ P is the difference in pressures inside and outside the liquid, 1/r the

2D curvature of the liquid surface and γ lg the liquid-air interfacial energy. Neglecting gravity ∆ P is constant everywhere across the liquid-air interface and therefore 1/r must be constant.

This implies that the curved liquid-air interfaces are circle (in fact cylinder) segments. As is schematized in figure 13, in this model a liquid along the edge of a crystal in contact with a substrate can appear in four different ways. The possibilities are i) a concave meniscus; ii) a convex meniscus; iii) a cylindrical, ‘droplet’-like configuration and iv) a ‘droplet’ only in contact with either the crystal or the substrate surface. In figure 13 also the transition states between the configurations are given, of which the straight meniscus, i.e. the state between i) and ii) will be the most relevant one. In the following part, we shall explore for which of the above cases the occurrence of a liquid layer stretching along the edge between crystal and substrate is energetically possible. For the concave and convex meniscus, i.e. case i) and ii), the condition for the occurrence of the stretched liquid is now given by:

OA ⋅ γ sl + OB ⋅ γ cl + arc ( AB ) ⋅ γ lg < OA ⋅ γ sg + OB ⋅ γ cg ,

(8)

with γ sl = substrate-liquid, γ sg = substrate-air, γ cl = crystal-liquid, γ cg = crystal-air, γ lg = liquid-air interfacial energies. OA, OB and arc(AB) are indicated in figures 13 i) and ii). Using


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the well-known Young relation for the contact angle θ of a liquid droplet on a substrate,

γ sl + γ lg cos θ sl = γ sg (and on the crystal, γ cl + γ lg cos θ cl = γ cg ) and some rewriting gives

OA ⋅ cos (θ sl ) + OB ⋅ cos(θ cl ) − arc( AB ) > 0

(9)

for both cases. Here θ sl and θ cl are the contact angles for the solution in contact with the sub-

Figure 13. Two-dimensional intersections of possible solution layer configurations at the corner between crystal and substrate. Only cases i) and i’) are stable and lead to crystal growth by TSC.



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strate and the crystal respectively. Some further geometric and goniometric analysis transforms equation (9) into the criterion

sin(θ cl +

α 2

) ⋅ cos(θ sl ) + sin(θ sl +

α 2

) ⋅ cosθ cl −

sin(ϕ )

α

⋅α > 0 ,

(10)

2 sin( ) 2

with α = π − ϕ − θ sl − θ cl , again for both cases. ϕ is the angle between the crystal side face and the substrate surface. It can be shown that criterion (10) is only fulfilled if

θ sl + θ cl + ϕ < π ,

or

arccos(

(11)

γ sg − γ sl γ − γ cl ) + arccos( cg ) +ϕ < π . γ lg γ lg

(11a)

Equation (11) implies that only for concave menisci a stretched liquid appears along the crystal-substrate edge and creeping is expected to occur. For convex menisci, the occurrence of a stretched liquid along the crystal-substrate edge is not favorable and no creeping will occur. The planar meniscus is the intermediate form between both cases. For the cylinder-like cases iii) and iv) it can be shown that also here the liquid is repelled towards the central droplet and again no creeping takes place. For case iv) one exception exists, namely if the crystal surface is completely wetted, i.e. if the contact angle θ cl = 0 . This happens if γ cg > γ cl + γ lg . Then the top faces as well as the side faces of the crystals are completely covered by a solution film, regardless of θsl.


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Creeping due to a complete wetting of the crystal surfaces was observed for KH2PO4 crystals growing on a glass substrate (section 3.2). The complete wetting of the KH2PO4 {100} surfaces was verified by placing a droplet of saturated aqueous solution on a prismatic face of a large KH2PO4 crystal, which was slightly etched by pure water to remove surface contamination. The droplet completely expanded and formed a liquid film with contact angle θ cl = 0 .

4.2. Bottom supplied creeping 4.2.1. Growth rate In the second important case of creeping, bottom supplied creeping (BSC), the supply of solution takes place via a narrow space between the growing crystallites and the substrate surface. BSC was sometimes found for K2Cr2O7, especially on rough substrate surfaces, and was always found for NH4Cl. It leads to planar crystallites with closed growth fronts. As shown in figure 12b, the solvent evaporates from a liquid front with a total thickness, Rf. Here we assume Rf to be equal to the sum of the liquid film thickness, ε, and the crystal height, h. Creeping is again determined by water vapor transport according to the stationary case of Fick’s law, ∇2c(r)=0. Now the problem is (quasi) two-dimensional and as an approximation it can be described in cylindrical coordinates [42]:

d 2 c 1 dc + = 0, dr 2 r dr

(12)

with r the distance from the linear growth front. Using the general solution c(r) = a ln(r) + b and the boundary conditions c(Rf) = cs and c(δ) = cb, with δ a kind of boundary layer thickness, one obtains [42]



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 c −c  r c( r ) = cb +  s b  ln .  ln( R / δ )  δ f  

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(13)

As δ >> Rf the influence of the liquid-solid contact angles and other details of the growth front can be neglected and the diffusion field can indeed be approximated as a hemi-cylinder. Using Fick’s first law and integrating over the hemi-cylinder gives the total water evaporation rate

 c − cb  , J tot = − Dπl  s  ln( R / δ )  f  

(14)

with l the length of the growing edge. Using Rf ≅ ε + h gives the propagation velocity of the creeping crystal front

vedge ≅

J tot s , hlρ

(15)

vedge ≅

Dsπ ( cb − cs ) . ρh ln((h + ε ) / δ )

(16)

or

This approximated equation again shows that the creeping velocity is proportional to the actual water vapor pressure, cb, minus the equilibrium pressure, cs ≈ ceq, and that it increases for thinner crystals (i.e. smaller h). Passing a flow of air reduces the boundary layer thickness, δ, and thus increases the propagation rate. This agrees with the observations. A somewhat different definition of the liquid front radius, Rf, does not alter the above conclusions.


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4.2.1. Wetting conditions Although the rate limiting factors for both TSC and BSC are qualitatively the same, the wetting condition for the occurrence of each form of creeping is essentially different. Figure 14 shows three possible configurations for the area, O, between the crystal and the substrate surface. The first case is direct contact between the crystal and the substrate giving a total interfacial energy Oγsc, with γsc the crystal-substrate interfacial energy. The second possibility is the occurrence of a narrow space filled with air corresponding to a total interfacial energy of O(γcg + γsg). Finally, this space can be filled with solution giving a total interface energy of O(γcl + γsl). A liquid layer underneath the creeping crystals, a necessary condition for BSC, only occurs if the third case is energetically most favorable, or if

γcl + γsl

{

< γsc .

(17)

< γcg + γsg

If this wetting requirement is fulfilled, BSC is especially favorable for thin, platy crystals, which, according to equation (16), give a maximal growth rate. Due to the closed growth front for such a crystal morphology, the aggregates are quite compact.

4.3. No or limited creeping No creeping occurs if the above mentioned wetting conditions for TSC and BSC are not fulfilled. In the case of CdI2 growth by anticreeping, the observed repulsion of the solution by the growing crystallites indicates that the crystal-solution contact angle,

θ cl = ar cos[( γ cg − γ cl ) / γ lg ] , is large. This increases the left-hand term of equation (11) and makes TSC less probable. As follows from equation (17), the increase of the crystal-solution interfacial energy, γcl, prevents BSC as well. In the case of ZnSO4 and BaCl2 growth, optical



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Figure 14.

Three possible configurations for the area between the crystal and the sub-

strate surface. Only the third case, i.e. U = O(γcl + γsl), leads to crystal growth by BSC.

microscopy showed that the liquid does not pass the narrow spaces between crystal and substrate or adjacent crystallites. Apparently, here the conditions for BSC and TSC are not fulfilled. The absence of creeping for K2Cr2O7 solution droplets on Teflon substrates is explained by the poor wetting of the solution to the substrate, giving a large value for

θ sl = arcos[( γ sg − γ sl ) / γ lg ] and γsl. From the criteria expressed in equations (11) and (17) it is obvious that this promotes neither TSC nor BSC. Because also in the cases where no creeping takes place, evaporation of solute from the droplet governs the crystallization process and because the solution layer is thinnest near the rim of the droplet, the supersaturation is maximal


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in this area. Therefore, the crystallites preferentially nucleate at the periphery of the droplet and then grow inwards. Instead of using a Teflon substrate to increase θ sl and γsl, a thin, water repelling film can be applied to a hydrophilic substrate. Application of such a layer on the side walls of a growth vessel is common practice to avoid unwanted creeping in crystal growth systems. During evaporation of KCl, potash alum, CuSO4 and (NH4)H2PO4 solution droplets very slow creeping was found, despite the fact that the conditions for TSC were clearly fulfilled. This is explained by the large thickness of the growing single crystals, which, according to equations (6) and (16), slows down creeping. The thickness of the crystals is caused by their more or less block shaped morphology as well as by the absence of secondary nucleation. The repeated generation of secondary nuclei leads to a repeated formation of small and therefore thin crystals at the growth front, which enhances creeping. Hence, not only the wetting behavior, but also secondary nucleation and crystal morphology should be considered in understanding the phenomenon of creeping.

5. Seaweed patterns

5.1. Pattern formation A growing body is morphologically unstable if its protruding parts grow faster than the rest of its surface. It is well known that this leads to many kinds of branched patterns as a result of the self-amplifying growth at the branch tips. For crystal growth the accelerated growth of protruding parts is a consequence of mass (heat) transport limitation, which is described by Fick’s (Fourier’s) laws. If the protruding parts of a dissolving body exhibit enhanced dissolution, then the body is morphologically stable and no ramified patterns develop. Hence, an evaporating droplet of liquid will not show branching. However, in the case of -31ACS Paragon Plus Environment


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creeping evaporation is accompanied by growth. Here the protruding droplet surface areas, which exhibit enhanced evaporation, show faster growth. Due to capillarity effects, the grown crystals “draw” solution from the central droplet and the “sticking-out” is reinforced. The continued sequence of solute evaporation, crystal growth and supply of solute by capillarity is a self-amplifying process, which can lead to morphological instability in the case of creeping. From a macroscopic point of view, all the patterns that were obtained by creeping were isotropic, regardless of anisotropic growth of the individual crystallites or the use of an anisotropic (single crystalline) substrate. No orientational order, i.e. no branching in preferred directions was found. This implies that all the branched patterns observed in this paper are seaweed type instead of dendritic. The main reason for the isotropic behavior is the fact that, at least for the systems described in this paper, the secondary nucleation is random as concerns crystal orientation. Dendritic patterns require oriented, i.e. epitaxial, secondary nucleation. Other, supplementary reasons for seaweed growth are the isotropic surface energy of the solution and the isotropic diffusion coefficient of the water vapor phase. Although the creeping patterns show no orientational order, they can still be compact or fractal. It is obvious that the closed NH4Cl pattern formed by BSC is compact, but the ramified patterns of K2Cr2O7 grown by TSC, need verification. Therefore for several K2Cr2O7 creeping patterns the Hausdorff dimension, DH, [29] was determined. This was realized by carefully selecting extended, fully grown branch-patterns not disturbed by neighboring patterns, which stem from one single crystallite emerging from the rim of the central droplet. These patterns were digitized using a CCD-camera and the surface area occupied by the crystallites, N(r), was plotted as a function of the distance, r, from the aggregate’s center of mass. A value of DH was then obtained using


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ln N ( r ) = D H ln( r ) .

(18)

Figure 15. Plot of ln(N(r)) versus ln(r) to determine the Hausdorff dimension of a K2Cr2O7 creeping pattern. N(r) is the surface area occupied by the grown crystallites, within a square with edge 2r. The square is centered on the aggregate’s center of mass.

From five graphs of the type shown in figure 15, it was found that DH = 2.00 ± 0.01. This integer value of the Hausdorff dimension means that the fractal-resembling K2Cr2O7 growth features in fact are compact.

4.2. Crystal chain formation and side branching The K2Cr2O7 creeping patterns are composed of branched chains of successive crystallites, which are formed by the repeated creation of secondary, 3D, nuclei of arbitrary crystallographic orientation near the tips of the growing crystallites. The absence of an orientational relationship suggests that the secondary nuclei are formed in the liquid in front of the crystal tips, where supersaturation is maximal, rather than on the tip surface itself. In-situ observation showed that the crystallite chains are not generated by a convective transport of small crystallites toward the aggregate tips.



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The average distance, , between the subsequently formed crystallites is determined by a competition between the tip growth rate, R, and the frequency of secondary nucleation, I, in the liquid volume ahead, according to

= R/I.

(19)

Side branching is initiated if instead of one, two nuclei are formed at the aggregate tip. Therefore, the frequency of side-branching is expected to increase for decreasing . From the high growth rates of several micrometers per second and the faceted morphologies observed during creeping it is very likely that tip growth proceeds by 2D nucleation according to the Birth and Spread mechanism [46]. Spiral growth is not expected for the fast growing tips, also because dislocations easily refract from the narrow top faces toward the large side faces during crystallite growth [47]. This gives a tip growth rate of

1 R ≅ v st exp( − ∆G2*D / kT ) , 3

(20)

with vst = β st ∆µ / kT the step velocity and ∆G2*D =

πΩcγ st2 the activation barrier for 2D nuhst ∆µ

cleation [46]. Here βst is the step kinetic coefficient, ∆µ/kT the driving force for crystal growth, Ωc the volume of one growth unit, hst the height of one growth unit and γst the edge free energy of a growth step. The frequency of secondary nucleation, I, in the liquid is given by [48-51]

(

)

I ≅ B ′ exp − ∆G3*D / kT ,

(21)


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with B′ a kinetic parameter. For (partially) spherical nuclei the activation barrier for 3D nucleation is given by ∆G3*D = f

16πγ cl3 Ω c 3(∆µ )

2

2

. For homogeneous nucleation in the liquid in front

of the tip f = 1; for heterogeneous nucleation in the liquid, on the substrate or previous crystallite surface f < 1. Combining equations (19), (20) and (21) gives an estimate for the average distance between subsequent crystallites

or

[

]

< d >=

vst exp − (∆G2*D / 3 − ∆G3*D ) / kT , B'

(22a)

< d >=

 1 vst exp − B'  kT

(22b)

2  πΩ cγ st2 16πγ cl3 Ω c    , − f 2  3h ∆µ 3(∆µ )   st

with ∆G3*D >> ∆G2*D / 3 . If growth does not take place too close to the roughening temperature, where γ st → 0 [52], then γ cl can be approximated by γ st / hst . From equation (22) it now follows that lowering the surface energy γcl (and thus also γst), leads to a decrease in . This explains why crystals with a strong tendency to faceting and thus a large γcl, such as NaCl and potash alum, form less easily crystal chains and side-branches. In addition, increasing the supersaturation ∆µ/kT lowers ∆G3*D more than ∆G2*D and thus enhances repeated nucleation and side-branching. This agrees with our in-situ observations of K2Cr2O7 creeping under a flow of dry nitrogen. Here, the increase in driving force leads to faster tip growth, smaller crystallites and increased side-branching. It should be realized that our model is highly oversimplified. In a few cases mechanical disturbances of the propagating liquid front, induced by large concentration gradients in combination with a free liquid surface [53] or as a result of repeated pinning of the liquid



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front by dirt particles on the substrate surface, were observed to trigger the formation of 3D nuclei. From this it follows that nucleation limited aggregation (NLA) as elaborated by Naiben Ming et al. [53], also plays some role in the process of side-branching.

5. Conclusions Although the experiments are relatively simple to carry out, a unified, quantitative description of the creeping phenomenon is virtually impossible. Based on a number of different experiments, important aspects of crystal growth by creeping and its morphological instability have been highlighted. From these analyses several conclusions are drawn.

i) Crystal growth by creeping proceeds by evaporation of solvent in front of the surfaces of the growing crystallites. Rate determining step in this process is the transport of solvent vapor in the ambient air. ii) Fresh solution, necessary to continue the creeping process is supplied by liquid flow along the growing crystallites. This can take place aside and on top of the growing crystallites (top supplied creeping, TSC) or in a narrow space between the crystallites and the substrate (bottom supplied creeping, BSC). iii) Criteria for the occurrence of TSC, BSC or no creeping at all, can be formulated in terms of the various interfacial energies, γij, involved and the angle of the crystallite side faces with the substrate. The shape and size of the growing crystallites play an important role in the rate of creeping. iv) The branched, seaweed patterns encountered in many cases of creeping result from the occurrence of secondary nucleation of crystallites combined with morphological instability. This morphological instability is introduced by crystal growth limited by solvent vapor transport. Dendritic, i.e. patterns with branches in preferred crystallographic directions were not found in this study. -36ACS Paragon Plus Environment

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v) The Hausdorff dimension of the seaweed patterns of K2Cr2O7 growing from aqueous solution droplets equals 2.00 ± 0.01, which corresponds to so-called- compact growth. In all cases, the creeping patterns do not show any orientational ordering. vi) The development and properties of branched crystallite chains in the creeping patterns are determined by the occurrence and frequency of secondary, 3D nucleation. This process is strongly dependent on the surface energy of the growing crystallites and the supersaturation of the solution near the growing tips.

In short, it can be stated that crystal growth by creeping, a quite common phenomenon in nature and laboratory, is a process of fascinating complexity, which involves many aspects of crystal growth science.



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[38] Deegan, R.D.; Bakajin, O.; Dupont, T.F.; Huber, G.; Nagel, S.R.; Witten T.A. Nature

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Rev. E 2000, 62, 756.

[40] Widjaja, E.; Harris, M.T. Computers and Chemical Engineering 2008, 32, 2169

[41] See also in: Evaporative Self-Assembly of Ordered Complex Structures, Lin, Z., Ed; World Scientific: Singapore, 2012.



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[42] See for instance: Incropera, F.P.; Dewitt, D.P.; Bergman, T.L.; Lavine, A.S. Fundamen-

tals of Heat and Mass Transport; John Wiley and Sohns, USA, 2006; 6th edition, pp. 894-898.

[43] Picknett, R.G.; Bexon, R. Journal of Colloid and Interface Science 1977, 61, 336.

[44] See for instance: Laidler, K.J.; Meiser; J.H. Physical Chemistry; Houghton Mifflin Company: Boston, 1995; p. 850.

[45] See for instance: Atkins, P.; de Paula, J. Physical Chemistry; Oxford University Press: Oxford, 2002; 7th edition, p. 151.

[46] Van der Eerden, J.P. In: Handbook of Crystal Growth; Hurle, D.T.J., Ed; Elsevier: Amsterdam, 1993; Vol 1, Chapter 6.

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[51] Kashchiev, D.; van Rosmalen, G.M. Cryst. Res. Technol. 2003, 38, 555.

[52] Bennema, P. In: Handbook of Crystal Growth; Hurle, D.T.J., Ed; Elsevier: Amsterdam, 1993; Vol 1, Chapter 7.

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For Table of Contents only On the Creeping of Saturated Salt Solutions Willem J.P. van Enckevort and J. H. Los

Synopsis Creeping is a well known, but annoying phenomenon in the preparation of crystals from solution, where growing crystallites gradually extend up the walls of the growth vessel through solution transport followed by solvent evaporation at the crystallite tips. Different creeping patters protruding from evaporating solution droplets are examined and interpreted using models involving ambient humidity, capillarity, nucleation and crystal shape.


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On the Creeping of Saturated Salt Solutions - Crystal Growth & Design

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