͑ Color ͒ Polymer crystallization morphologies as a function of polymer composition. The relative clay polymer mass has been fixed at 5% ͑ OM images rendered in false color ͒ . ͑ a ͒ Spherulitic crystallization of a film of pure PEO ( T m ϭ 340 K, ␦ T ϭ 0.09). ͑ b ͒ Seaweed dendritic growth in a ͑ 50/50 ͒ PEO-PMMA film ( T m ϭ 332 K, ␦ T ϭ 0.07). ͑ c ͒ Symmetric dendritic growth in a ͑ 30/70 ͒ PEO-PMMA film. ͑ d ͒ 

Contexts in source publication

Context 1

... 4. ͑ Color ͒ Kinetics of polymer dendritic growth. ͑ a ͒ The dendrite perimeter p vs area A determines an apparent fractal dimension of the growing dendrites. The data points denote observations based on a figure of the growth of the dendrite shown in Fig. 2 ͑ a ͒ . The inset denotes the average tip velocity for the morphologies shown in Fig. 1 for a fixed degree of undercooling, ␦ T ϭ 0.07 ͑ ᭹ , pure PEO spherulites; ᭿ , 60:40 spherulites; ᭡ , 50/50 seaweed dendrite; छ , 30/70 symmetric dendrite ͒ . ͑ b ͒ The average rate of crystallization for ͑ 30/70 ͒ symmetric dendrite L ϭ 160 nm films as a function of undercooling ⌬ T ϭ ( T m Ϫ T c ).  ...

Context 2

... separation morphology that presumably formed during the film drying ͓ 38 ͔ and we find a smoothing of this surface topographical structure for temperatures above a temperature-composition locus that resembles a UCST cloud point curve ͓ 38 ͔ . By studying the temperature dependence of this smoothing we estimated the UCST critical composition ␾ c and critical temperature, ␾ c Ϸ 0.55 and T c Ϸ 378 K ͓ 38 ͔ . Previous determinations of cloud point curves of relatively thick PEO/PMMA polymer films cast from chloroform indicate T c ϭ 365 K and ␾ c ϭ 0.53 for a PEO-PMMA blend having molecular masses of M w PEO ϭ 4 ϫ 10 4 and M w PMMA ϭ 10 5 g mol Ϫ 1 ͓ 32 ͔ and T c ϭ 381 K and ␾ c ϭ 0.62 for molecular masses of M w PEO ϭ 10 6 and M w PMMA ϭ 10 6 g mol Ϫ 1 ͓ 32,39 ͔ . It should be appreciated that the residual solvent in the spun-cast film probably influences the phase separation be- havior that we observe and we can also expect the solvent to ‘‘plasticize’’ the film i.e., modify the dynamics of the film related to the glass transition of PMMA, leading to increased molecular mobility in these viscous films ͒ . Although the residual solvent effect complicates the determination of temperature range where phase separation occurs, it should not change the qualitative nature of the crystallization phenomenon under investigation. The thermodynamics and general aspects of phase separation and crystallization in monotectic mixtures ͑ two liquids and one solid phase ͒ are discussed by Cahn ͓ 40 ͔ . Reflective optical images were obtained with an optical microscope ͑ OM ͒ using a Nikon optical microscope ͓ 32 ͔ with a digital Kodak MegaPlus, charge-coupled device cam- era attachment ͓ 32 ͔ . We follow the growth kinetics of the patterns using automated data acquisition with a resolution of 1024 ϫ 1024 pixels. All the atomic force microscopy ͑ AFM ͒ experiments were carried out in air by using a Dimension 3100 microscope from Digital Instruments operating in the Tapping modeTM ͓ 32 ͔ . In this mode, the cantilever is forced to oscillate at a frequency close to its resonance frequency with an adjustable amplitude. The tip, attached to the cantilever, was a pure silicon single crystal tip ͑ model TSEP ͒ with a radius of curvature of about 10 nm. The tip contacts briefly the film surface at each low position of the cantilever and the amplitude of the oscillation varies. ‘‘Height’’ images are obtained by using the feedback loop that keeps the amplitude at a constant value by translating vertically the sample with the pi- ezoelectric scanner: height measurements are deduced from those displacements. For the engagement we used a ratio A sp / A 0 ϭ 0.9, where A 0 is the free oscillation amplitude and A sp the set-point one ͓ 41 ͔ . The (512 ϫ 512 pixels) images have been obtained by using a (100 ϫ 100 ␮ m 2 ) piezoelec- tric scanner; the scanning frequency was 0.5 Hz and the mean value of the repulsive normal force was 0.1 nN. All the ‘‘height’’images have been filtered through the ‘‘Planefit’’ procedure ͓ 41 ͔ . We also used the AFM to measure the thickness of the film scratching the surface or masking a border of the wafer before spin casting the solution. The vertical resolution of AFM is 0.1 Å. In an earlier work, we investigated the real space structure of ͑ amorphous ͒ polymer-blend phase separation by forming nearly two-dimensional polymer films in which one of the polymer components segregates to both the solid substrate and the polymer-air boundary ͓ 30 ͔ . These ‘‘ultrathin’’ films were also restricted to film thickness range in which phase separation occurs within the plane of the blend film ͓ 30 ͔ . The difference in the surface tension between the blend components causes the film to buckle in response to phase separation within the film ͓ 30 ͔ . This buckling provides a good source of contrast in optical and AFM measurements, en- abling high-resolution measurements of the dynamics of the ordering process in real space. Here we extend this earlier work by incorporating a model crystallizable polymer ͑ PEO ͒ into the film to study dendritic growth. We mix PEO with an amorphous polymer ͑ PMMA ͒ , which is the component that segregates to the film boundaries. Clay particles are added as nucleating centers for the PEO crystallization. The clay particles are convenient because they induce centrosymmetric crystallization patterns. Crystallization can also be induced by scratching or piercing the film with a sharp implement so these particles are not essential for inducing crystallization. The crystallization morphology of PEO mixed with PMMA in a thin-film geometry can be ‘‘tuned’’ through spherulitic, seaweed, symmetric dendritic, and fractal dendritic patterns through the adjustment of the PMMA composition. These crystallization morphologies are described in a separate paper ͓ 31 ͔ and here we briefly review the essential nature of this phenomenon before specializing our discussion to the growth of symmetric dendrites. In Fig. 1 we illustrate changes in the crystallization morphology of PEO/PMMA blend films arising from a variation in the polymer composition. Crystallization was performed at 305 K and the clay concentration of the spin-casting solution was fixed at 5% of the mass of the blend. The PEO melting temperature T m depends on the polymer composition and T m values are indicated in the caption, along with the undercooling, ␦ T ϭ ( T m Ϫ T crys )/ T m . Clay particles can be seen at the center of the patterns shown in Fig. 1, confirming that the clay acts as a nucleating agent. Over a large range of PEO mass fraction ͑ 50–100 % of PEO by mass ͒ , we find circularly symmetric spherulites ͓ Fig. 1 ͑ b ͔͒ . This is the ‘‘normal’’ polymer crystallization morphology encountered under processing conditions ͓ 1,2,14 –16 ͔ . In the insert of Fig. 1 ͑ a ͒ , we show the late-stage spherulitic crystallization morphology where the spherulites impinge on each other and deform to form a do- main wall morphology similar in appearance to a Voronoi cell pattern ͓ 42 ͔ . The sidebranching of the spherulite ‘‘needles’’ becomes increasingly coarse with the increasing PMMA composition in this concentration regime ͓ 31 ͔ , but the spherulites tend to retain their nearly circular shape. At an almost 50/50 polymer blend ͑ PEO/PMMA ͒ mass ratio, we find a regime ͓ Fig. 1 ͑ b ͔͒ where the spherulite morphology changes into a seaweed dendrite morphology ͓ 19 ͔ . This morphology exhibits broad growing tips that split intermittently and one of the newly formed branches normally grows to predominate over the other. The dominant branch ͑ ‘‘alpha branch’’ ͒ then splits again and the process repeats itself. Tip splitting is the dominant feature of seaweed dendritic growth and this morphology is well known from recent modeling of nonequilibrium crystallization ͑ see below ͒ and has been confirmed in many experimental studies. Increasing the PMMA concentration further to near 30/70 leads to another dramatic change in the polymer crystallization morphology. In Fig. 1 ͑ c ͒ , we observe well-formed symmetric dendrites where the fourfold symmetry of equilibrium PEO crystallization asserts itself at a macroscopic scale ͓ 43 ͔ . The solution and the melt grown crystals have the same crystal structure ͑ square- shaped crystals ͒ under near-equilibrium conditions ͓ 43 ͔ . ͑ We have observed nearly square crystals in our films when we crystallize at 331 K near T m . Note the near registry of the sidebranches on each side of the growing dendrite arm and the uniformity of the ‘‘starlike’’ envelope curve describing the positions of the sidebranch tips of the dendrite. Symmetric dendritic polymer crystallization patterns have often been observed in polymer crystals grown on surfaces from polymer solutions ͓ 44 ͔ , but we are unaware of previous observations of SDC in melt blends. ͑ However, distorted spherulitic and randomly branched crystallization morphologies have been observed in melt blends ͓ 45 ͔ . ͒ At still higher concentrations of PMMA ͑ 20/80 ͒ , we observe another morphological transition from the symmetric dendritic crystallization to the highly branched, fractal morphology illustrated in Fig. 1 ͑ d ͒ ͓ 31 ͔ . This interesting transition is not well understood yet, but we do note that the high concentration of PMMA makes the film highly viscous and this could have an impact on the stability of the dendrite tips and the gross crystallization morphology. It is also notable that the growth should be diffusion-limited in this regime and the low concentration of PEO could also contribute to the noisy nature of the resulting crystal growth in this regime. In the following, we focus specifically on the dynamics of symmetric dendrite crystallization, corresponding to relative PEO/PMMA-mass concentration of 30/70 and 5% clay by mass, relative to the polymer. Figures 2 a –2 d show optical images false color of the growth of a ͑ PEO-rich ͒ symmetric dendrite over a sequence of times from 60 to 460 min. The film thickness is 160 nm and the dimensionless undercooling T ( T m T c )/ T m equals 0.01. Note the cusplike shape of the envelope curve describing the tip positions of the dendrite arms, a feature observed previously in symmetric dendrite growth at relatively high undercooling ͓ 46 ͔ . The sidebranches of the dendrite in Fig. 2 ͑ a ͒ grow nearly perpendicularly to the slender and nearly parabolic main branch of the dendrite. Our dendritic crystallization images were acquired at a rate of one picture every 5 min and in Fig. 2 ͑ b ͒ we show the increase in the tip position from the center of the dendrite. ͑ The clay seed at the dendrite center is the noticeable dark spot at the center of the dendrite. ͒ We observe that the tip position of the dendrite grows in an oscillatory manner about an average constant rate R o ϭ 0.171 ␮ m/min. The period P of the tip growth oscillations is of the order of 100 min. It is important to realize that the dendrite ...

Context 3

... at a frequency close to its resonance frequency with an adjustable amplitude. The tip, attached to the cantilever, was a pure silicon single crystal tip ͑ model TSEP ͒ with a radius of curvature of about 10 nm. The tip contacts briefly the film surface at each low position of the cantilever and the amplitude of the oscillation varies. ‘‘Height’’ images are obtained by using the feedback loop that keeps the amplitude at a constant value by translating vertically the sample with the pi- ezoelectric scanner: height measurements are deduced from those displacements. For the engagement we used a ratio A sp / A 0 ϭ 0.9, where A 0 is the free oscillation amplitude and A sp the set-point one ͓ 41 ͔ . The (512 ϫ 512 pixels) images have been obtained by using a (100 ϫ 100 ␮ m 2 ) piezoelec- tric scanner; the scanning frequency was 0.5 Hz and the mean value of the repulsive normal force was 0.1 nN. All the ‘‘height’’images have been filtered through the ‘‘Planefit’’ procedure ͓ 41 ͔ . We also used the AFM to measure the thickness of the film scratching the surface or masking a border of the wafer before spin casting the solution. The vertical resolution of AFM is 0.1 Å. In an earlier work, we investigated the real space structure of ͑ amorphous ͒ polymer-blend phase separation by forming nearly two-dimensional polymer films in which one of the polymer components segregates to both the solid substrate and the polymer-air boundary ͓ 30 ͔ . These ‘‘ultrathin’’ films were also restricted to film thickness range in which phase separation occurs within the plane of the blend film ͓ 30 ͔ . The difference in the surface tension between the blend components causes the film to buckle in response to phase separation within the film ͓ 30 ͔ . This buckling provides a good source of contrast in optical and AFM measurements, en- abling high-resolution measurements of the dynamics of the ordering process in real space. Here we extend this earlier work by incorporating a model crystallizable polymer ͑ PEO ͒ into the film to study dendritic growth. We mix PEO with an amorphous polymer ͑ PMMA ͒ , which is the component that segregates to the film boundaries. Clay particles are added as nucleating centers for the PEO crystallization. The clay particles are convenient because they induce centrosymmetric crystallization patterns. Crystallization can also be induced by scratching or piercing the film with a sharp implement so these particles are not essential for inducing crystallization. The crystallization morphology of PEO mixed with PMMA in a thin-film geometry can be ‘‘tuned’’ through spherulitic, seaweed, symmetric dendritic, and fractal dendritic patterns through the adjustment of the PMMA composition. These crystallization morphologies are described in a separate paper ͓ 31 ͔ and here we briefly review the essential nature of this phenomenon before specializing our discussion to the growth of symmetric dendrites. In Fig. 1 we illustrate changes in the crystallization morphology of PEO/PMMA blend films arising from a variation in the polymer composition. Crystallization was performed at 305 K and the clay concentration of the spin-casting solution was fixed at 5% of the mass of the blend. The PEO melting temperature T m depends on the polymer composition and T m values are indicated in the caption, along with the undercooling, ␦ T ϭ ( T m Ϫ T crys )/ T m . Clay particles can be seen at the center of the patterns shown in Fig. 1, confirming that the clay acts as a nucleating agent. Over a large range of PEO mass fraction ͑ 50–100 % of PEO by mass ͒ , we find circularly symmetric spherulites ͓ Fig. 1 ͑ b ͔͒ . This is the ‘‘normal’’ polymer crystallization morphology encountered under processing conditions ͓ 1,2,14 –16 ͔ . In the insert of Fig. 1 ͑ a ͒ , we show the late-stage spherulitic crystallization morphology where the spherulites impinge on each other and deform to form a do- main wall morphology similar in appearance to a Voronoi cell pattern ͓ 42 ͔ . The sidebranching of the spherulite ‘‘needles’’ becomes increasingly coarse with the increasing PMMA composition in this concentration regime ͓ 31 ͔ , but the spherulites tend to retain their nearly circular shape. At an almost 50/50 polymer blend ͑ PEO/PMMA ͒ mass ratio, we find a regime ͓ Fig. 1 ͑ b ͔͒ where the spherulite morphology changes into a seaweed dendrite morphology ͓ 19 ͔ . This morphology exhibits broad growing tips that split intermittently and one of the newly formed branches normally grows to predominate over the other. The dominant branch ͑ ‘‘alpha branch’’ ͒ then splits again and the process repeats itself. Tip splitting is the dominant feature of seaweed dendritic growth and this morphology is well known from recent modeling of nonequilibrium crystallization ͑ see below ͒ and has been confirmed in many experimental studies. Increasing the PMMA concentration further to near 30/70 leads to another dramatic change in the polymer crystallization morphology. In Fig. 1 ͑ c ͒ , we observe well-formed symmetric dendrites where the fourfold symmetry of equilibrium PEO crystallization asserts itself at a macroscopic scale ͓ 43 ͔ . The solution and the melt grown crystals have the same crystal structure ͑ square- shaped crystals ͒ under near-equilibrium conditions ͓ 43 ͔ . ͑ We have observed nearly square crystals in our films when we crystallize at 331 K near T m . Note the near registry of the sidebranches on each side of the growing dendrite arm and the uniformity of the ‘‘starlike’’ envelope curve describing the positions of the sidebranch tips of the dendrite. Symmetric dendritic polymer crystallization patterns have often been observed in polymer crystals grown on surfaces from polymer solutions ͓ 44 ͔ , but we are unaware of previous observations of SDC in melt blends. ͑ However, distorted spherulitic and randomly branched crystallization morphologies have been observed in melt blends ͓ 45 ͔ . ͒ At still higher concentrations of PMMA ͑ 20/80 ͒ , we observe another morphological transition from the symmetric dendritic crystallization to the highly branched, fractal morphology illustrated in Fig. 1 ͑ d ͒ ͓ 31 ͔ . This interesting transition is not well understood yet, but we do note that the high concentration of PMMA makes the film highly viscous and this could have an impact on the stability of the dendrite tips and the gross crystallization morphology. It is also notable that the growth should be diffusion-limited in this regime and the low concentration of PEO could also contribute to the noisy nature of the resulting crystal growth in this regime. In the following, we focus specifically on the dynamics of symmetric dendrite crystallization, corresponding to relative PEO/PMMA-mass concentration of 30/70 and 5% clay by mass, relative to the polymer. Figures 2 a –2 d show optical images false color of the growth of a ͑ PEO-rich ͒ symmetric dendrite over a sequence of times from 60 to 460 min. The film thickness is 160 nm and the dimensionless undercooling T ( T m T c )/ T m equals 0.01. Note the cusplike shape of the envelope curve describing the tip positions of the dendrite arms, a feature observed previously in symmetric dendrite growth at relatively high undercooling ͓ 46 ͔ . The sidebranches of the dendrite in Fig. 2 ͑ a ͒ grow nearly perpendicularly to the slender and nearly parabolic main branch of the dendrite. Our dendritic crystallization images were acquired at a rate of one picture every 5 min and in Fig. 2 ͑ b ͒ we show the increase in the tip position from the center of the dendrite. ͑ The clay seed at the dendrite center is the noticeable dark spot at the center of the dendrite. ͒ We observe that the tip position of the dendrite grows in an oscillatory manner about an average constant rate R o ϭ 0.171 ␮ m/min. The period P of the tip growth oscillations is of the order of 100 min. It is important to realize that the dendrite morphology changes in thinner films. Figure 2 ͑ c ͒ shows an example of dendritic growth in a 50-nm-thick film, where the crystallization conditions ͑ temperature, composition ͒ are the same as in Fig. 2 ͑ a ͒ . ͑ This morphological transition is discussed below. ͒ Notably, the dendrite in Fig. 2 ͑ c ͒ does not exhibit growth pulsations and has a more disordered appearance and its boundary envelope has a squarelike shape. The inset in Fig. 2 ͑ c ͒ shows an AFM image of the dendrite tip region, showing again a similarity to the form in the AFM and optical images. Despite differences in the large-scale crystallization morphology, the tip radius in Fig. 2 ͑ c ͒ is nearly the same as in Fig. 2 ͑ a ͒ , ( r Ϸ 1 ␮ m). The AFM data is discussed quantitatively below. We obtain some insight into these dendritic growth patterns by comparing to phase field simulations of two- dimensional SDC in a two-dimensional fluid mixture ͓ 47 ͔ . The simulation in Fig. 2 ͑ d ͒ corresponds to a Ni-Cu alloy ( ␾ Ni ϭ 0.59), where ␧ is taken to have a relatively large value, ␧ ϭ 0.05 and ␦ T is relatively large for metallurgical fluids, ␦ T ϭ 0.013. „ The new phase field calculation in Fig. 2 ͑ d ͒ is for ␧ ϭ 0.05, which is larger than the ␧ considered in previous work ͓ 47 ͔ ( ␧ ϭ 0.04), but otherwise the model parameters are identical to those specified in Ref. ͓ 47 ͔ . ... Com- parison of the simulation to our measurements is meant to be only qualitative. The main point is that growth pulsations are not observed in the two-dimensional phase field simulation, but we do find a reasonable resemblance between the ‘‘two- dimensional’’ polymer dendritic growth shown in Fig. 2 ͑ c ͒ and the simulated crystallization patterns ͓ Fig. 2 ͑ d ͔͒ . Apparently, no ␧ measurements have ever been made on high molecular weight polymers, and at present we are restricted to qualitative comparisons between the phase field model and our measurements. The oscillations in the tip radius in Fig. 2 ͑ b ͒ are quantified by subtracting the average dendrite tip radius R ( t ...

Context 4

... indicate T c ϭ 365 K and ␾ c ϭ 0.53 for a PEO-PMMA blend having molecular masses of M w PEO ϭ 4 ϫ 10 4 and M w PMMA ϭ 10 5 g mol Ϫ 1 ͓ 32 ͔ and T c ϭ 381 K and ␾ c ϭ 0.62 for molecular masses of M w PEO ϭ 10 6 and M w PMMA ϭ 10 6 g mol Ϫ 1 ͓ 32,39 ͔ . It should be appreciated that the residual solvent in the spun-cast film probably influences the phase separation be- havior that we observe and we can also expect the solvent to ‘‘plasticize’’ the film i.e., modify the dynamics of the film related to the glass transition of PMMA, leading to increased molecular mobility in these viscous films ͒ . Although the residual solvent effect complicates the determination of temperature range where phase separation occurs, it should not change the qualitative nature of the crystallization phenomenon under investigation. The thermodynamics and general aspects of phase separation and crystallization in monotectic mixtures ͑ two liquids and one solid phase ͒ are discussed by Cahn ͓ 40 ͔ . Reflective optical images were obtained with an optical microscope ͑ OM ͒ using a Nikon optical microscope ͓ 32 ͔ with a digital Kodak MegaPlus, charge-coupled device cam- era attachment ͓ 32 ͔ . We follow the growth kinetics of the patterns using automated data acquisition with a resolution of 1024 ϫ 1024 pixels. All the atomic force microscopy ͑ AFM ͒ experiments were carried out in air by using a Dimension 3100 microscope from Digital Instruments operating in the Tapping modeTM ͓ 32 ͔ . In this mode, the cantilever is forced to oscillate at a frequency close to its resonance frequency with an adjustable amplitude. The tip, attached to the cantilever, was a pure silicon single crystal tip ͑ model TSEP ͒ with a radius of curvature of about 10 nm. The tip contacts briefly the film surface at each low position of the cantilever and the amplitude of the oscillation varies. ‘‘Height’’ images are obtained by using the feedback loop that keeps the amplitude at a constant value by translating vertically the sample with the pi- ezoelectric scanner: height measurements are deduced from those displacements. For the engagement we used a ratio A sp / A 0 ϭ 0.9, where A 0 is the free oscillation amplitude and A sp the set-point one ͓ 41 ͔ . The (512 ϫ 512 pixels) images have been obtained by using a (100 ϫ 100 ␮ m 2 ) piezoelec- tric scanner; the scanning frequency was 0.5 Hz and the mean value of the repulsive normal force was 0.1 nN. All the ‘‘height’’images have been filtered through the ‘‘Planefit’’ procedure ͓ 41 ͔ . We also used the AFM to measure the thickness of the film scratching the surface or masking a border of the wafer before spin casting the solution. The vertical resolution of AFM is 0.1 Å. In an earlier work, we investigated the real space structure of ͑ amorphous ͒ polymer-blend phase separation by forming nearly two-dimensional polymer films in which one of the polymer components segregates to both the solid substrate and the polymer-air boundary ͓ 30 ͔ . These ‘‘ultrathin’’ films were also restricted to film thickness range in which phase separation occurs within the plane of the blend film ͓ 30 ͔ . The difference in the surface tension between the blend components causes the film to buckle in response to phase separation within the film ͓ 30 ͔ . This buckling provides a good source of contrast in optical and AFM measurements, en- abling high-resolution measurements of the dynamics of the ordering process in real space. Here we extend this earlier work by incorporating a model crystallizable polymer ͑ PEO ͒ into the film to study dendritic growth. We mix PEO with an amorphous polymer ͑ PMMA ͒ , which is the component that segregates to the film boundaries. Clay particles are added as nucleating centers for the PEO crystallization. The clay particles are convenient because they induce centrosymmetric crystallization patterns. Crystallization can also be induced by scratching or piercing the film with a sharp implement so these particles are not essential for inducing crystallization. The crystallization morphology of PEO mixed with PMMA in a thin-film geometry can be ‘‘tuned’’ through spherulitic, seaweed, symmetric dendritic, and fractal dendritic patterns through the adjustment of the PMMA composition. These crystallization morphologies are described in a separate paper ͓ 31 ͔ and here we briefly review the essential nature of this phenomenon before specializing our discussion to the growth of symmetric dendrites. In Fig. 1 we illustrate changes in the crystallization morphology of PEO/PMMA blend films arising from a variation in the polymer composition. Crystallization was performed at 305 K and the clay concentration of the spin-casting solution was fixed at 5% of the mass of the blend. The PEO melting temperature T m depends on the polymer composition and T m values are indicated in the caption, along with the undercooling, ␦ T ϭ ( T m Ϫ T crys )/ T m . Clay particles can be seen at the center of the patterns shown in Fig. 1, confirming that the clay acts as a nucleating agent. Over a large range of PEO mass fraction ͑ 50–100 % of PEO by mass ͒ , we find circularly symmetric spherulites ͓ Fig. 1 ͑ b ͔͒ . This is the ‘‘normal’’ polymer crystallization morphology encountered under processing conditions ͓ 1,2,14 –16 ͔ . In the insert of Fig. 1 ͑ a ͒ , we show the late-stage spherulitic crystallization morphology where the spherulites impinge on each other and deform to form a do- main wall morphology similar in appearance to a Voronoi cell pattern ͓ 42 ͔ . The sidebranching of the spherulite ‘‘needles’’ becomes increasingly coarse with the increasing PMMA composition in this concentration regime ͓ 31 ͔ , but the spherulites tend to retain their nearly circular shape. At an almost 50/50 polymer blend ͑ PEO/PMMA ͒ mass ratio, we find a regime ͓ Fig. 1 ͑ b ͔͒ where the spherulite morphology changes into a seaweed dendrite morphology ͓ 19 ͔ . This morphology exhibits broad growing tips that split intermittently and one of the newly formed branches normally grows to predominate over the other. The dominant branch ͑ ‘‘alpha branch’’ ͒ then splits again and the process repeats itself. Tip splitting is the dominant feature of seaweed dendritic growth and this morphology is well known from recent modeling of nonequilibrium crystallization ͑ see below ͒ and has been confirmed in many experimental studies. Increasing the PMMA concentration further to near 30/70 leads to another dramatic change in the polymer crystallization morphology. In Fig. 1 ͑ c ͒ , we observe well-formed symmetric dendrites where the fourfold symmetry of equilibrium PEO crystallization asserts itself at a macroscopic scale ͓ 43 ͔ . The solution and the melt grown crystals have the same crystal structure ͑ square- shaped crystals ͒ under near-equilibrium conditions ͓ 43 ͔ . ͑ We have observed nearly square crystals in our films when we crystallize at 331 K near T m . Note the near registry of the sidebranches on each side of the growing dendrite arm and the uniformity of the ‘‘starlike’’ envelope curve describing the positions of the sidebranch tips of the dendrite. Symmetric dendritic polymer crystallization patterns have often been observed in polymer crystals grown on surfaces from polymer solutions ͓ 44 ͔ , but we are unaware of previous observations of SDC in melt blends. ͑ However, distorted spherulitic and randomly branched crystallization morphologies have been observed in melt blends ͓ 45 ͔ . ͒ At still higher concentrations of PMMA ͑ 20/80 ͒ , we observe another morphological transition from the symmetric dendritic crystallization to the highly branched, fractal morphology illustrated in Fig. 1 ͑ d ͒ ͓ 31 ͔ . This interesting transition is not well understood yet, but we do note that the high concentration of PMMA makes the film highly viscous and this could have an impact on the stability of the dendrite tips and the gross crystallization morphology. It is also notable that the growth should be diffusion-limited in this regime and the low concentration of PEO could also contribute to the noisy nature of the resulting crystal growth in this regime. In the following, we focus specifically on the dynamics of symmetric dendrite crystallization, corresponding to relative PEO/PMMA-mass concentration of 30/70 and 5% clay by mass, relative to the polymer. Figures 2 a –2 d show optical images false color of the growth of a ͑ PEO-rich ͒ symmetric dendrite over a sequence of times from 60 to 460 min. The film thickness is 160 nm and the dimensionless undercooling T ( T m T c )/ T m equals 0.01. Note the cusplike shape of the envelope curve describing the tip positions of the dendrite arms, a feature observed previously in symmetric dendrite growth at relatively high undercooling ͓ 46 ͔ . The sidebranches of the dendrite in Fig. 2 ͑ a ͒ grow nearly perpendicularly to the slender and nearly parabolic main branch of the dendrite. Our dendritic crystallization images were acquired at a rate of one picture every 5 min and in Fig. 2 ͑ b ͒ we show the increase in the tip position from the center of the dendrite. ͑ The clay seed at the dendrite center is the noticeable dark spot at the center of the dendrite. ͒ We observe that the tip position of the dendrite grows in an oscillatory manner about an average constant rate R o ϭ 0.171 ␮ m/min. The period P of the tip growth oscillations is of the order of 100 min. It is important to realize that the dendrite morphology changes in thinner films. Figure 2 ͑ c ͒ shows an example of dendritic growth in a 50-nm-thick film, where the crystallization conditions ͑ temperature, composition ͒ are the same as in Fig. 2 ͑ a ͒ . ͑ This morphological transition is discussed below. ͒ Notably, the dendrite in Fig. 2 ͑ c ͒ does not exhibit growth pulsations and has a more disordered appearance and its boundary envelope has a squarelike shape. The inset in Fig. 2 ͑ c ͒ shows an AFM image of the dendrite tip region, showing again a ...

Context 5

... the dendrite sidebranch width w ( t ) we choose an arbitrary sidebranch ͓ denoted by arrow in Fig. 2 ͑ a ͔͒ and define w ( t ) as the orthogonal distance from the tip of the sidebranch to the center line of the main dendrite arm. The sidebranch width grows with an average rate ͑ equal to R o to within experimental uncertainty ͒ with oscillations ␦ w ( t ) about this average. Our determination of ␦ w ( t ) in Fig. 3 ͑ a ͒ shows that ␦ w ( t ) oscillates out of phase with ␦ R ( t ). Further, a ‘‘phase plot’’ of ␦ R ( t ) versus ␦ w ( t ) in Fig. 3 ͑ b ͒ reveals that the dendritic growth in Fig. 3 ͑ a ͒ is governed by a limit cycle with a phase angle ␣ difference of about 164° ͑ see caption of Fig. 3 ͒ . The dendritic growth in Fig. 2 ͑ a ͒ has a self-similar appearance and this suggests that it might be useful to determine the apparent mass-scaling dimension ͑ fractal dimension ͒ describing the symmetric polymer dendritic growth. The determination of fractal dimension d f was determined from image analysis based on the area-perimeter technique ͓ 48 ͔ . In Fig. 4 ͑ a ͒ , we show a plot of the polymer dendrite area A and perimeter p obtained by digitizing the optical image series corresponding to the dendrite growth shown in Fig. 1 ͑ a ͒ . We observe that a power law relation A ϳ p 1/ d f , can fit fairly well the data and we determine d f from the slope of log p versus log A . This gives an apparent fractal dimension d f ϭ 1.78 Ϯ 0.05, where the correlation coefficient for the power law fit is R 2 ϭ 0.98. Notably, we do not observe growth pulsations in Fig. 4 ͑ a ͒ so that A and p 1/1.78 must exhibit similar growth oscillations. The near linearity of the plot in Fig. 4 ͑ a ͒ confirms the impression of the near self- similarity of the polymer dendritic growth pattern. The temperature dependence of the rate of dendritic growth for a L ϭ 160 nm film is shown in Fig. 4 ͑ b ͒ as a function of undercooling, ⌬ T ϭ ( T m Ϫ T c ). Over the temperature range investigated, the rate of crystallization R 0 can be described by a power law, R 0 ϳ ( ⌬ T ) ␦ where ␦ Ϸ 2.53 Ϯ 0.02. The correlation coefficient for the power law fit is R 2 ϭ 0.99. A power scaling of R 0 with an effective exponent near 2.6 has been suggested to be a ‘‘universal’’ property of dendritic growth in small molecule liquids ͓ 49 ͔ . Since theory offers limited guidance about the factors governing the period P of dendrite growth pulsations, we explore the influence of some obvious system parameters under our control—undercooling, polymer composition ͑ supersaturation ͒ and film thickness, L . In Fig. 5 ͑ a ͒ we show ␦ R ( t ) for a range of undercooling ⌬ T ϭ ( T m Ϫ T c ) values in the range ͑ 288 –308 K ͒ . Apparently, P has no appreciable dependence on ⌬ T , but Fig. 5 ͑ b ͒ shows that ␦ A R increases nearly exponentially with ⌬ T . The correlation coefficient for the exponential fit is R 2 ϭ 0.99. A change in the relative polymer composition has a large influence on the pulsation rate, but this effect can cause a qualitative change in the crystallization morphology ͓ 31 ͔ so we restrict ourselves to a composition range where the SDC is observed. A decrease of the PMMA concentration causes a decrease in p and an increase in ␦ A R for a fixed ⌬ T ϭ 35 K, ͑ e.g., P Ϸ 105 min and ␦ A R Ϸ 15 ␮ m for a 35/65 blend while P Ϸ 180 min and ␦ A R Ϸ 9 ␮ m for a 30/70 blend ͒ . We next explore the change in the growth pattern dynamics and morphology associated with reducing the film’s thickness. Figure 6 shows that P first increases sharply with decreasing film thickness L , but then drops precipitously to zero below a critical film thickness, L c Ϸ 80 nm. The SDC is similar to Fig. 1 ͑ a ͒ for L Ͼ L c , but we observe a different morphology for L Ͻ L c ͑ see Fig. 2 ͑ c ͒ and inset to Fig. 6 ͒ . Thus, we have direct evidence that the morphological transition is accompanied by a change in the dynamics of the dendrite tip. The lack of pulsations in the ‘‘two-dimensional’’ blend film dendrites is also reflected in the extent of correlation in the position of the sidebranches on each side of the primary growing parabolic dendrite arms ͓ see Fig. 2 ͑ c ͔͒ . The registry of sidebranches and the cusplike envelope curve describing the positions of the sidebranch tips in the symmetric dendrite shown in Fig. 2 ͑ a ͒ are contrasted with the ‘‘two- dimensional’’ dendrite ( L Ͻ L c ) shown in Fig. 2 ͑ c ͒ . There is little correlation in the sidebranch positions on either side of this dendrite. This enhanced regularity of structure in the pulsing dendrite is reminiscent of the regular sidebranching found in the growth of dendritic growth subjected to periodic external perturbations ͓ 50,51 ͔ . We, therefore, suggest that the oscillatory tip mode imparts regularity to the growing dendrite. Finally, we should mention in this section that oscillatory growth front modes have recently been reported in the spherulite and seaweed crystallization morphologies for other materials ͑ Fig. 1; see Discussion ͒ so that the presence of hydrodynamic modes in propagating crystallization fronts appears to be a general, but nonuniversal, phenomenon. However, the study of the dynamics of these other nonequilibrium crystallization morphologies will require the devel- opment of specialized measurement techniques for each morphology ͑ the splitting of the dendrite tip creates some ambiguity in defining the precise location of the crystallization front ͒ , so in the present paper, we confine our attention to symmetric dendrite growth. The singular nature of the shift in P with L provides an important clue into the nature of the dendritic growth pulsa- tions. At first, we anticipated that P would correspond to a diffusion-controlled depletion time that would scale quadrati- cally with film thickness. This expectation would lead to a decrease of P with film thickness; an effect opposite to our measurements. We then realized that the L dependence of P is similar to the finite-size dependence of pulsations observed in oscillatory chemical reactions. This comparison is natural because Belousov-Zhabotinsky ͑ BZ ͒ reactions also exhibit pattern formation with propagating wave fronts. The oscillation period of the BZ reaction occurring in ion- exchange beads ͓ 52 ͔ ͑ which causes the color of the beads to flicker ͒ likewise increases strongly with decreasing bead radius and the oscillations cease when the bead size became smaller than a critical radius ͑ 0.2 mm ͒ . In Fig. 6, we compare our measurements of P to the functional form suggested by the studies of finite-size effects on the BZ reaction ͑ bead radius is replaced by polymer film thickness ͒ ͓ 52 ͔ . This leads to the relation, P ϭ P ρ /(1 Ϫ L / L c ) for L Ͼ L c , P ρ ϭ 90 min and P ϭ 0 for L Ͻ L c . The correlation coefficient for the data point fit is R 2 ϭ 0.99. The finite-size dependence of the oscillation period in the BZ reaction was attributed in Ref. ͓ 51 ͔ to a change in the reaction rate due to the inactive nature of the reaction at the bead surface, leading to a correction of the reaction rate involving the surface-volume ratio. In our own measurements, the boundaries of the blend film are enriched in PMMA so that a similar finite-size effect on the pulsation period is plausible. The viewpoint of a supercooled liquid as a variety of the excitable medium and crystallization as a variety of reaction- diffusion wave propagation also gives insight into the influence of the clay particles on the crystallization morphology. At low concentrations, the clay particles mainly serve as centers of the dendritic growth and similar dendritic patterns can be obtained by punching or scratching the film without clay. The ‘‘catalyst’’ particles play a similar role as an excitation source for BZ reactions in solutions loaded with ferroin- loaded resin beads ͓ 53 ͔ . As is well known, the BZ reaction in a fluid layer gives rise to symmetric ‘‘target’’ chemical waves at low bead concentrations, but these patterns break up into rotating spiral patterns at higher bead concentrations due to the interference between the chemical waves ͓ 53–55 ͔ . Our proposed analogy between nonequilibrium crystallization and autocatalytic chemical reactions would lead us to expect a similar symmetry breaking phenomenon in dendritic ...

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... resolution of AFM is 0.1 Å. In an earlier work, we investigated the real space structure of ͑ amorphous ͒ polymer-blend phase separation by forming nearly two-dimensional polymer films in which one of the polymer components segregates to both the solid substrate and the polymer-air boundary ͓ 30 ͔ . These ‘‘ultrathin’’ films were also restricted to film thickness range in which phase separation occurs within the plane of the blend film ͓ 30 ͔ . The difference in the surface tension between the blend components causes the film to buckle in response to phase separation within the film ͓ 30 ͔ . This buckling provides a good source of contrast in optical and AFM measurements, en- abling high-resolution measurements of the dynamics of the ordering process in real space. Here we extend this earlier work by incorporating a model crystallizable polymer ͑ PEO ͒ into the film to study dendritic growth. We mix PEO with an amorphous polymer ͑ PMMA ͒ , which is the component that segregates to the film boundaries. Clay particles are added as nucleating centers for the PEO crystallization. The clay particles are convenient because they induce centrosymmetric crystallization patterns. Crystallization can also be induced by scratching or piercing the film with a sharp implement so these particles are not essential for inducing crystallization. The crystallization morphology of PEO mixed with PMMA in a thin-film geometry can be ‘‘tuned’’ through spherulitic, seaweed, symmetric dendritic, and fractal dendritic patterns through the adjustment of the PMMA composition. These crystallization morphologies are described in a separate paper ͓ 31 ͔ and here we briefly review the essential nature of this phenomenon before specializing our discussion to the growth of symmetric dendrites. In Fig. 1 we illustrate changes in the crystallization morphology of PEO/PMMA blend films arising from a variation in the polymer composition. Crystallization was performed at 305 K and the clay concentration of the spin-casting solution was fixed at 5% of the mass of the blend. The PEO melting temperature T m depends on the polymer composition and T m values are indicated in the caption, along with the undercooling, ␦ T ϭ ( T m Ϫ T crys )/ T m . Clay particles can be seen at the center of the patterns shown in Fig. 1, confirming that the clay acts as a nucleating agent. Over a large range of PEO mass fraction ͑ 50–100 % of PEO by mass ͒ , we find circularly symmetric spherulites ͓ Fig. 1 ͑ b ͔͒ . This is the ‘‘normal’’ polymer crystallization morphology encountered under processing conditions ͓ 1,2,14 –16 ͔ . In the insert of Fig. 1 ͑ a ͒ , we show the late-stage spherulitic crystallization morphology where the spherulites impinge on each other and deform to form a do- main wall morphology similar in appearance to a Voronoi cell pattern ͓ 42 ͔ . The sidebranching of the spherulite ‘‘needles’’ becomes increasingly coarse with the increasing PMMA composition in this concentration regime ͓ 31 ͔ , but the spherulites tend to retain their nearly circular shape. At an almost 50/50 polymer blend ͑ PEO/PMMA ͒ mass ratio, we find a regime ͓ Fig. 1 ͑ b ͔͒ where the spherulite morphology changes into a seaweed dendrite morphology ͓ 19 ͔ . This morphology exhibits broad growing tips that split intermittently and one of the newly formed branches normally grows to predominate over the other. The dominant branch ͑ ‘‘alpha branch’’ ͒ then splits again and the process repeats itself. Tip splitting is the dominant feature of seaweed dendritic growth and this morphology is well known from recent modeling of nonequilibrium crystallization ͑ see below ͒ and has been confirmed in many experimental studies. Increasing the PMMA concentration further to near 30/70 leads to another dramatic change in the polymer crystallization morphology. In Fig. 1 ͑ c ͒ , we observe well-formed symmetric dendrites where the fourfold symmetry of equilibrium PEO crystallization asserts itself at a macroscopic scale ͓ 43 ͔ . The solution and the melt grown crystals have the same crystal structure ͑ square- shaped crystals ͒ under near-equilibrium conditions ͓ 43 ͔ . ͑ We have observed nearly square crystals in our films when we crystallize at 331 K near T m . Note the near registry of the sidebranches on each side of the growing dendrite arm and the uniformity of the ‘‘starlike’’ envelope curve describing the positions of the sidebranch tips of the dendrite. Symmetric dendritic polymer crystallization patterns have often been observed in polymer crystals grown on surfaces from polymer solutions ͓ 44 ͔ , but we are unaware of previous observations of SDC in melt blends. ͑ However, distorted spherulitic and randomly branched crystallization morphologies have been observed in melt blends ͓ 45 ͔ . ͒ At still higher concentrations of PMMA ͑ 20/80 ͒ , we observe another morphological transition from the symmetric dendritic crystallization to the highly branched, fractal morphology illustrated in Fig. 1 ͑ d ͒ ͓ 31 ͔ . This interesting transition is not well understood yet, but we do note that the high concentration of PMMA makes the film highly viscous and this could have an impact on the stability of the dendrite tips and the gross crystallization morphology. It is also notable that the growth should be diffusion-limited in this regime and the low concentration of PEO could also contribute to the noisy nature of the resulting crystal growth in this regime. In the following, we focus specifically on the dynamics of symmetric dendrite crystallization, corresponding to relative PEO/PMMA-mass concentration of 30/70 and 5% clay by mass, relative to the polymer. Figures 2 a –2 d show optical images false color of the growth of a ͑ PEO-rich ͒ symmetric dendrite over a sequence of times from 60 to 460 min. The film thickness is 160 nm and the dimensionless undercooling T ( T m T c )/ T m equals 0.01. Note the cusplike shape of the envelope curve describing the tip positions of the dendrite arms, a feature observed previously in symmetric dendrite growth at relatively high undercooling ͓ 46 ͔ . The sidebranches of the dendrite in Fig. 2 ͑ a ͒ grow nearly perpendicularly to the slender and nearly parabolic main branch of the dendrite. Our dendritic crystallization images were acquired at a rate of one picture every 5 min and in Fig. 2 ͑ b ͒ we show the increase in the tip position from the center of the dendrite. ͑ The clay seed at the dendrite center is the noticeable dark spot at the center of the dendrite. ͒ We observe that the tip position of the dendrite grows in an oscillatory manner about an average constant rate R o ϭ 0.171 ␮ m/min. The period P of the tip growth oscillations is of the order of 100 min. It is important to realize that the dendrite morphology changes in thinner films. Figure 2 ͑ c ͒ shows an example of dendritic growth in a 50-nm-thick film, where the crystallization conditions ͑ temperature, composition ͒ are the same as in Fig. 2 ͑ a ͒ . ͑ This morphological transition is discussed below. ͒ Notably, the dendrite in Fig. 2 ͑ c ͒ does not exhibit growth pulsations and has a more disordered appearance and its boundary envelope has a squarelike shape. The inset in Fig. 2 ͑ c ͒ shows an AFM image of the dendrite tip region, showing again a similarity to the form in the AFM and optical images. Despite differences in the large-scale crystallization morphology, the tip radius in Fig. 2 ͑ c ͒ is nearly the same as in Fig. 2 ͑ a ͒ , ( r Ϸ 1 ␮ m). The AFM data is discussed quantitatively below. We obtain some insight into these dendritic growth patterns by comparing to phase field simulations of two- dimensional SDC in a two-dimensional fluid mixture ͓ 47 ͔ . The simulation in Fig. 2 ͑ d ͒ corresponds to a Ni-Cu alloy ( ␾ Ni ϭ 0.59), where ␧ is taken to have a relatively large value, ␧ ϭ 0.05 and ␦ T is relatively large for metallurgical fluids, ␦ T ϭ 0.013. „ The new phase field calculation in Fig. 2 ͑ d ͒ is for ␧ ϭ 0.05, which is larger than the ␧ considered in previous work ͓ 47 ͔ ( ␧ ϭ 0.04), but otherwise the model parameters are identical to those specified in Ref. ͓ 47 ͔ . ... Com- parison of the simulation to our measurements is meant to be only qualitative. The main point is that growth pulsations are not observed in the two-dimensional phase field simulation, but we do find a reasonable resemblance between the ‘‘two- dimensional’’ polymer dendritic growth shown in Fig. 2 ͑ c ͒ and the simulated crystallization patterns ͓ Fig. 2 ͑ d ͔͒ . Apparently, no ␧ measurements have ever been made on high molecular weight polymers, and at present we are restricted to qualitative comparisons between the phase field model and our measurements. The oscillations in the tip radius in Fig. 2 ͑ b ͒ are quantified by subtracting the average dendrite tip radius R ( t ) ϭ R 0 t ͓ straight line in Fig. 2 ͑ b ͔͒ from R ( t ). Figure 3 ͑ a ͒ shows that the tip position fluctuation ␦ R ( t ) ϭ R ( t ) Ϫ R ( t ) is nearly sinusoidal; the solid curves correspond to a fit of ␦ R ( t ) to ␦ A R sin(2 ␲ t / P ), where ␦ A R is the oscillation amplitude and P is the pulsation period . We next compare ␦ R ( t ) to a measure of fluctuations in the width ␦ w ( t ) of the dendrite arm. To determine the dendrite sidebranch width w ( t ) we choose an arbitrary sidebranch ͓ denoted by arrow in Fig. 2 ͑ a ͔͒ and define w ( t ) as the orthogonal distance from the tip of the sidebranch to the center line of the main dendrite arm. The sidebranch width grows with an average rate ͑ equal to R o to within experimental uncertainty ͒ with oscillations ␦ w ( t ) about this average. Our determination of ␦ w ( t ) in Fig. 3 ͑ a ͒ shows that ␦ w ( t ) oscillates out of phase with ␦ R ( t ). Further, a ‘‘phase plot’’ of ␦ R ( t ) versus ␦ w ( t ) in Fig. 3 ͑ b ͒ reveals that the dendritic growth in Fig. 3 ͑ a ͒ is governed by a limit cycle with a phase angle ␣ difference of about 164° ͑ see caption of Fig. 3 ͒ . The dendritic ...

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... findings contradicted an early theoretical treatment of dendritic growth ͑ ‘‘geometric model’’ of crystallization front movement ͒ ͓ 23,24 ͔ that predicted the possibility of oscillatory tip growth, but later models did not yield growth pulsations ͓ 24 ͔ . The observation of growth pulsations in the ‘‘geometric model’’ is restricted to values of the surface tension anisotropy ␧ close to a critical value ␧ c , where the symmetric dendrites first form, and the model further predicted that the growth oscillations damp to zero for larger values of ␧ ͓ 23 ͔ . This model of crystal growth then implies that the observation of pulsating dendritic growth in symmetric dendrites should depend on a particular value of ␧ . Since ␧ is normally nearly independent of temperature for a pure material, it is difficult to draw general conclusions about the presence of growth pulsations based on the observation of the crystallization of particular substances. Never- theless, recent theoretical work ͓ 25,26 ͔ has emphasized the view, supported by the experimental findings mentioned before, that the amplification of thermal noise is generally the source of sidebranch growth in symmetric dendritic crystallization. Spontaneous and coherent sidebranching in directional solidification has recently been observed in both experiment and simulation ͓ 27–29 ͔ and these observations suggest that oscillatory hydrodynamic modes of the dendrite tip can provide an alternative source sidebranch generation in dendritic growth ͓ 27–29 ͔ . The present paper examines the nature of growth pulsations in SDC crystallization in polymer-blend films where the crystallization rate is much lower than the values normally found in small molecule liquids and metal alloys ͓ 1 ͔ . Our crystallization measurements correspond to free dendritic growth rather than directional solidification and to a geometry that is nearly two-dimensional. The use of polymeric fluids to slow down the dynamics of ordering has been exploited in real-space studies of polymer- blend phase separation ͓ 30 ͔ and this approach allows high resolution measurements of the dynamics of nonequilibrium polymer crystallization using optical microscopy and atomic force microscopy. In the present work, we confine ourselves to investigating the growth of symmetric dendrites in thin polymer-blend films under relatively large undercooling conditions. We employ a polymeric blend of a crystallizable polymer ͓ polyethylene oxide ͑ PEO ͔͒ and amorphous polymer ͓ polymethyl methacrylate ͑ PMMA ͔͒ that allows us to tune the surface tension anisotropy ␧ and, thus, the qualitative crystallization morphology ͓ 31 ͔ . This system should al- low us a better chance of observing tip growth pulsations and other dynamical phenomena that might occur near certain critical values of ␧ , where there are transitions between dendritic growth morphologies. PMMA and PEO materials were purchased from Aldrich ͓ 32 ͔ and their polydispersity indices k ( k ϭ M w / M n ) were determined at NIST by gel permeation chromatography to equal k (PMMA) ϭ 1.8 ͑ M w ϭ 7.3 ϫ 10 3 g mol Ϫ 1 ͓ 32 ͔͒ and k (PEO)( M w ϭ 1.5 ϫ 10 5 g mol Ϫ 1 ͓ 32 ͔͒ Ϸ 4. The equilibrium melting temperature T m of pure PEO was determined to equal T m ϭ 338 K by differential-scanning calorimetry on thick ͑ 20 ␮ m ͒ evaporated PEO/chloroform films and the glass transition temperatures of the PEO and PMMA films were found to equal T g ϭ 213 and 377 K, respectively. Our estimate of T m for PEO agrees well with previously reported values ͓ 33 ͔ . Montmorillonite, ‘‘Cloisite’’ ͑ MON ͒ , was sup- plied by Southern Clay Products ͓ 32 ͔ . This clay mineral has exchangeable sodium ions, and a cation exchanged capacity of ca. 120 meq per 100 g. One gram of MON and 50 ml of distilled water at 353 K were placed in 100-ml beaker along with 1 g of distearyldimethyl ammonium chloride. The mixture was stirred vigorously for 1 h, and then it was filtered and washed three times with 100 ml of hot water to remove NaCl. After being washed with ethanol ͑ 50 ml ͒ to remove any excess of ammonium salt, the product was freeze dried, and kept in a vacuum oven at room temperature for 24 h. The resulting organically modified montmorillonite ͑ OMON ͒ dis- persed well in chloroform, although the unmodified MON did not do disperse well in the polymer-blend spin-casting solution. The blend components were dissolved in chloroform at a concentration between 0.3% and 3% relative weight of the polymer to solvent, unless otherwise stated. Thin blend films of this solution were then spin coated onto Si substrates ͓ Semiconductor Processing Co. ͓ 32 ͔ , orientation ͑ 100 ͒ , Type P ͔ at a spin speed of 2000 rpm. This procedure results in films of uniform thickness between 100 and 500 nm. Isother- mal crystallization was made by heating the films at 377 K for 10 min ͑ above the melting temperature ͒ and then cooled down quickly ͑ 50 K/min ͒ to the desired crystallization temperature ͑ see caption of Fig. 1 for specifications of crystallization temperatures in our measurements ͒ . Prior to spin coat- ing, the polished Si substrates were treated for 2 h with a solution of 70% H 2 SO 4 /30% H 2 O 2 at 353 K and then rinsed with de-ionized water. There have been several previous studies of blends of PEO and PMMA, encompassing mixtures of components of various molecular weights, and this blend is usually indicated to be miscible over a wide temperature range ͓ 33–35 ͔ . However, lower critical solution temperature ͑ LCST ͒ phase separation has been reported in PEO-PMMA films for temperatures below the critical temperature of about 623 K ͓ 36,37 ͔ . The as-cast films have the appearance of a phase separation morphology that presumably formed during the film drying ͓ 38 ͔ and we find a smoothing of this surface topographical structure for temperatures above a temperature-composition locus that resembles a UCST cloud point curve ͓ 38 ͔ . By studying the temperature dependence of this smoothing we estimated the UCST critical composition ␾ c and critical temperature, ␾ c Ϸ 0.55 and T c Ϸ 378 K ͓ 38 ͔ . Previous determinations of cloud point curves of relatively thick PEO/PMMA polymer films cast from chloroform indicate T c ϭ 365 K and ␾ c ϭ 0.53 for a PEO-PMMA blend having molecular masses of M w PEO ϭ 4 ϫ 10 4 and M w PMMA ϭ 10 5 g mol Ϫ 1 ͓ 32 ͔ and T c ϭ 381 K and ␾ c ϭ 0.62 for molecular masses of M w PEO ϭ 10 6 and M w PMMA ϭ 10 6 g mol Ϫ 1 ͓ 32,39 ͔ . It should be appreciated that the residual solvent in the spun-cast film probably influences the phase separation be- havior that we observe and we can also expect the solvent to ‘‘plasticize’’ the film i.e., modify the dynamics of the film related to the glass transition of PMMA, leading to increased molecular mobility in these viscous films ͒ . Although the residual solvent effect complicates the determination of temperature range where phase separation occurs, it should not change the qualitative nature of the crystallization phenomenon under investigation. The thermodynamics and general aspects of phase separation and crystallization in monotectic mixtures ͑ two liquids and one solid phase ͒ are discussed by Cahn ͓ 40 ͔ . Reflective optical images were obtained with an optical microscope ͑ OM ͒ using a Nikon optical microscope ͓ 32 ͔ with a digital Kodak MegaPlus, charge-coupled device cam- era attachment ͓ 32 ͔ . We follow the growth kinetics of the patterns using automated data acquisition with a resolution of 1024 ϫ 1024 pixels. All the atomic force microscopy ͑ AFM ͒ experiments were carried out in air by using a Dimension 3100 microscope from Digital Instruments operating in the Tapping modeTM ͓ 32 ͔ . In this mode, the cantilever is forced to oscillate at a frequency close to its resonance frequency with an adjustable amplitude. The tip, attached to the cantilever, was a pure silicon single crystal tip ͑ model TSEP ͒ with a radius of curvature of about 10 nm. The tip contacts briefly the film surface at each low position of the cantilever and the amplitude of the oscillation varies. ‘‘Height’’ images are obtained by using the feedback loop that keeps the amplitude at a constant value by translating vertically the sample with the pi- ezoelectric scanner: height measurements are deduced from those displacements. For the engagement we used a ratio A sp / A 0 ϭ 0.9, where A 0 is the free oscillation amplitude and A sp the set-point one ͓ 41 ͔ . The (512 ϫ 512 pixels) images have been obtained by using a (100 ϫ 100 ␮ m 2 ) piezoelec- tric scanner; the scanning frequency was 0.5 Hz and the mean value of the repulsive normal force was 0.1 nN. All the ‘‘height’’images have been filtered through the ‘‘Planefit’’ procedure ͓ 41 ͔ . We also used the AFM to measure the thickness of the film scratching the surface or masking a border of the wafer before spin casting the solution. The vertical resolution of AFM is 0.1 Å. In an earlier work, we investigated the real space structure of ͑ amorphous ͒ polymer-blend phase separation by forming nearly two-dimensional polymer films in which one of the polymer components segregates to both the solid substrate and the polymer-air boundary ͓ 30 ͔ . These ‘‘ultrathin’’ films were also restricted to film thickness range in which phase separation occurs within the plane of the blend film ͓ 30 ͔ . The difference in the surface tension between the blend components causes the film to buckle in response to phase separation within the film ͓ 30 ͔ . This buckling provides a good source of contrast in optical and AFM measurements, en- abling high-resolution measurements of the dynamics of the ordering process in real space. Here we extend this earlier work by incorporating a model crystallizable polymer ͑ PEO ͒ into the film to study dendritic growth. We mix PEO with an amorphous polymer ͑ PMMA ͒ , which is the component that segregates to the film boundaries. Clay particles are added as nucleating centers for the PEO ...

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... film probably influences the phase separation be- havior that we observe and we can also expect the solvent to ‘‘plasticize’’ the film i.e., modify the dynamics of the film related to the glass transition of PMMA, leading to increased molecular mobility in these viscous films ͒ . Although the residual solvent effect complicates the determination of temperature range where phase separation occurs, it should not change the qualitative nature of the crystallization phenomenon under investigation. The thermodynamics and general aspects of phase separation and crystallization in monotectic mixtures ͑ two liquids and one solid phase ͒ are discussed by Cahn ͓ 40 ͔ . Reflective optical images were obtained with an optical microscope ͑ OM ͒ using a Nikon optical microscope ͓ 32 ͔ with a digital Kodak MegaPlus, charge-coupled device cam- era attachment ͓ 32 ͔ . We follow the growth kinetics of the patterns using automated data acquisition with a resolution of 1024 ϫ 1024 pixels. All the atomic force microscopy ͑ AFM ͒ experiments were carried out in air by using a Dimension 3100 microscope from Digital Instruments operating in the Tapping modeTM ͓ 32 ͔ . In this mode, the cantilever is forced to oscillate at a frequency close to its resonance frequency with an adjustable amplitude. The tip, attached to the cantilever, was a pure silicon single crystal tip ͑ model TSEP ͒ with a radius of curvature of about 10 nm. The tip contacts briefly the film surface at each low position of the cantilever and the amplitude of the oscillation varies. ‘‘Height’’ images are obtained by using the feedback loop that keeps the amplitude at a constant value by translating vertically the sample with the pi- ezoelectric scanner: height measurements are deduced from those displacements. For the engagement we used a ratio A sp / A 0 ϭ 0.9, where A 0 is the free oscillation amplitude and A sp the set-point one ͓ 41 ͔ . The (512 ϫ 512 pixels) images have been obtained by using a (100 ϫ 100 ␮ m 2 ) piezoelec- tric scanner; the scanning frequency was 0.5 Hz and the mean value of the repulsive normal force was 0.1 nN. All the ‘‘height’’images have been filtered through the ‘‘Planefit’’ procedure ͓ 41 ͔ . We also used the AFM to measure the thickness of the film scratching the surface or masking a border of the wafer before spin casting the solution. The vertical resolution of AFM is 0.1 Å. In an earlier work, we investigated the real space structure of ͑ amorphous ͒ polymer-blend phase separation by forming nearly two-dimensional polymer films in which one of the polymer components segregates to both the solid substrate and the polymer-air boundary ͓ 30 ͔ . These ‘‘ultrathin’’ films were also restricted to film thickness range in which phase separation occurs within the plane of the blend film ͓ 30 ͔ . The difference in the surface tension between the blend components causes the film to buckle in response to phase separation within the film ͓ 30 ͔ . This buckling provides a good source of contrast in optical and AFM measurements, en- abling high-resolution measurements of the dynamics of the ordering process in real space. Here we extend this earlier work by incorporating a model crystallizable polymer ͑ PEO ͒ into the film to study dendritic growth. We mix PEO with an amorphous polymer ͑ PMMA ͒ , which is the component that segregates to the film boundaries. Clay particles are added as nucleating centers for the PEO crystallization. The clay particles are convenient because they induce centrosymmetric crystallization patterns. Crystallization can also be induced by scratching or piercing the film with a sharp implement so these particles are not essential for inducing crystallization. The crystallization morphology of PEO mixed with PMMA in a thin-film geometry can be ‘‘tuned’’ through spherulitic, seaweed, symmetric dendritic, and fractal dendritic patterns through the adjustment of the PMMA composition. These crystallization morphologies are described in a separate paper ͓ 31 ͔ and here we briefly review the essential nature of this phenomenon before specializing our discussion to the growth of symmetric dendrites. In Fig. 1 we illustrate changes in the crystallization morphology of PEO/PMMA blend films arising from a variation in the polymer composition. Crystallization was performed at 305 K and the clay concentration of the spin-casting solution was fixed at 5% of the mass of the blend. The PEO melting temperature T m depends on the polymer composition and T m values are indicated in the caption, along with the undercooling, ␦ T ϭ ( T m Ϫ T crys )/ T m . Clay particles can be seen at the center of the patterns shown in Fig. 1, confirming that the clay acts as a nucleating agent. Over a large range of PEO mass fraction ͑ 50–100 % of PEO by mass ͒ , we find circularly symmetric spherulites ͓ Fig. 1 ͑ b ͔͒ . This is the ‘‘normal’’ polymer crystallization morphology encountered under processing conditions ͓ 1,2,14 –16 ͔ . In the insert of Fig. 1 ͑ a ͒ , we show the late-stage spherulitic crystallization morphology where the spherulites impinge on each other and deform to form a do- main wall morphology similar in appearance to a Voronoi cell pattern ͓ 42 ͔ . The sidebranching of the spherulite ‘‘needles’’ becomes increasingly coarse with the increasing PMMA composition in this concentration regime ͓ 31 ͔ , but the spherulites tend to retain their nearly circular shape. At an almost 50/50 polymer blend ͑ PEO/PMMA ͒ mass ratio, we find a regime ͓ Fig. 1 ͑ b ͔͒ where the spherulite morphology changes into a seaweed dendrite morphology ͓ 19 ͔ . This morphology exhibits broad growing tips that split intermittently and one of the newly formed branches normally grows to predominate over the other. The dominant branch ͑ ‘‘alpha branch’’ ͒ then splits again and the process repeats itself. Tip splitting is the dominant feature of seaweed dendritic growth and this morphology is well known from recent modeling of nonequilibrium crystallization ͑ see below ͒ and has been confirmed in many experimental studies. Increasing the PMMA concentration further to near 30/70 leads to another dramatic change in the polymer crystallization morphology. In Fig. 1 ͑ c ͒ , we observe well-formed symmetric dendrites where the fourfold symmetry of equilibrium PEO crystallization asserts itself at a macroscopic scale ͓ 43 ͔ . The solution and the melt grown crystals have the same crystal structure ͑ square- shaped crystals ͒ under near-equilibrium conditions ͓ 43 ͔ . ͑ We have observed nearly square crystals in our films when we crystallize at 331 K near T m . Note the near registry of the sidebranches on each side of the growing dendrite arm and the uniformity of the ‘‘starlike’’ envelope curve describing the positions of the sidebranch tips of the dendrite. Symmetric dendritic polymer crystallization patterns have often been observed in polymer crystals grown on surfaces from polymer solutions ͓ 44 ͔ , but we are unaware of previous observations of SDC in melt blends. ͑ However, distorted spherulitic and randomly branched crystallization morphologies have been observed in melt blends ͓ 45 ͔ . ͒ At still higher concentrations of PMMA ͑ 20/80 ͒ , we observe another morphological transition from the symmetric dendritic crystallization to the highly branched, fractal morphology illustrated in Fig. 1 ͑ d ͒ ͓ 31 ͔ . This interesting transition is not well understood yet, but we do note that the high concentration of PMMA makes the film highly viscous and this could have an impact on the stability of the dendrite tips and the gross crystallization morphology. It is also notable that the growth should be diffusion-limited in this regime and the low concentration of PEO could also contribute to the noisy nature of the resulting crystal growth in this regime. In the following, we focus specifically on the dynamics of symmetric dendrite crystallization, corresponding to relative PEO/PMMA-mass concentration of 30/70 and 5% clay by mass, relative to the polymer. Figures 2 a –2 d show optical images false color of the growth of a ͑ PEO-rich ͒ symmetric dendrite over a sequence of times from 60 to 460 min. The film thickness is 160 nm and the dimensionless undercooling T ( T m T c )/ T m equals 0.01. Note the cusplike shape of the envelope curve describing the tip positions of the dendrite arms, a feature observed previously in symmetric dendrite growth at relatively high undercooling ͓ 46 ͔ . The sidebranches of the dendrite in Fig. 2 ͑ a ͒ grow nearly perpendicularly to the slender and nearly parabolic main branch of the dendrite. Our dendritic crystallization images were acquired at a rate of one picture every 5 min and in Fig. 2 ͑ b ͒ we show the increase in the tip position from the center of the dendrite. ͑ The clay seed at the dendrite center is the noticeable dark spot at the center of the dendrite. ͒ We observe that the tip position of the dendrite grows in an oscillatory manner about an average constant rate R o ϭ 0.171 ␮ m/min. The period P of the tip growth oscillations is of the order of 100 min. It is important to realize that the dendrite morphology changes in thinner films. Figure 2 ͑ c ͒ shows an example of dendritic growth in a 50-nm-thick film, where the crystallization conditions ͑ temperature, composition ͒ are the same as in Fig. 2 ͑ a ͒ . ͑ This morphological transition is discussed below. ͒ Notably, the dendrite in Fig. 2 ͑ c ͒ does not exhibit growth pulsations and has a more disordered appearance and its boundary envelope has a squarelike shape. The inset in Fig. 2 ͑ c ͒ shows an AFM image of the dendrite tip region, showing again a similarity to the form in the AFM and optical images. Despite differences in the large-scale crystallization morphology, the tip radius in Fig. 2 ͑ c ͒ is nearly the same as in Fig. 2 ͑ a ͒ , ( r Ϸ 1 ␮ m). The AFM data is discussed quantitatively below. We obtain some insight into these dendritic growth patterns by comparing ...

Context 9

... for molecular masses of M w PEO ϭ 10 6 and M w PMMA ϭ 10 6 g mol Ϫ 1 ͓ 32,39 ͔ . It should be appreciated that the residual solvent in the spun-cast film probably influences the phase separation be- havior that we observe and we can also expect the solvent to ‘‘plasticize’’ the film i.e., modify the dynamics of the film related to the glass transition of PMMA, leading to increased molecular mobility in these viscous films ͒ . Although the residual solvent effect complicates the determination of temperature range where phase separation occurs, it should not change the qualitative nature of the crystallization phenomenon under investigation. The thermodynamics and general aspects of phase separation and crystallization in monotectic mixtures ͑ two liquids and one solid phase ͒ are discussed by Cahn ͓ 40 ͔ . Reflective optical images were obtained with an optical microscope ͑ OM ͒ using a Nikon optical microscope ͓ 32 ͔ with a digital Kodak MegaPlus, charge-coupled device cam- era attachment ͓ 32 ͔ . We follow the growth kinetics of the patterns using automated data acquisition with a resolution of 1024 ϫ 1024 pixels. All the atomic force microscopy ͑ AFM ͒ experiments were carried out in air by using a Dimension 3100 microscope from Digital Instruments operating in the Tapping modeTM ͓ 32 ͔ . In this mode, the cantilever is forced to oscillate at a frequency close to its resonance frequency with an adjustable amplitude. The tip, attached to the cantilever, was a pure silicon single crystal tip ͑ model TSEP ͒ with a radius of curvature of about 10 nm. The tip contacts briefly the film surface at each low position of the cantilever and the amplitude of the oscillation varies. ‘‘Height’’ images are obtained by using the feedback loop that keeps the amplitude at a constant value by translating vertically the sample with the pi- ezoelectric scanner: height measurements are deduced from those displacements. For the engagement we used a ratio A sp / A 0 ϭ 0.9, where A 0 is the free oscillation amplitude and A sp the set-point one ͓ 41 ͔ . The (512 ϫ 512 pixels) images have been obtained by using a (100 ϫ 100 ␮ m 2 ) piezoelec- tric scanner; the scanning frequency was 0.5 Hz and the mean value of the repulsive normal force was 0.1 nN. All the ‘‘height’’images have been filtered through the ‘‘Planefit’’ procedure ͓ 41 ͔ . We also used the AFM to measure the thickness of the film scratching the surface or masking a border of the wafer before spin casting the solution. The vertical resolution of AFM is 0.1 Å. In an earlier work, we investigated the real space structure of ͑ amorphous ͒ polymer-blend phase separation by forming nearly two-dimensional polymer films in which one of the polymer components segregates to both the solid substrate and the polymer-air boundary ͓ 30 ͔ . These ‘‘ultrathin’’ films were also restricted to film thickness range in which phase separation occurs within the plane of the blend film ͓ 30 ͔ . The difference in the surface tension between the blend components causes the film to buckle in response to phase separation within the film ͓ 30 ͔ . This buckling provides a good source of contrast in optical and AFM measurements, en- abling high-resolution measurements of the dynamics of the ordering process in real space. Here we extend this earlier work by incorporating a model crystallizable polymer ͑ PEO ͒ into the film to study dendritic growth. We mix PEO with an amorphous polymer ͑ PMMA ͒ , which is the component that segregates to the film boundaries. Clay particles are added as nucleating centers for the PEO crystallization. The clay particles are convenient because they induce centrosymmetric crystallization patterns. Crystallization can also be induced by scratching or piercing the film with a sharp implement so these particles are not essential for inducing crystallization. The crystallization morphology of PEO mixed with PMMA in a thin-film geometry can be ‘‘tuned’’ through spherulitic, seaweed, symmetric dendritic, and fractal dendritic patterns through the adjustment of the PMMA composition. These crystallization morphologies are described in a separate paper ͓ 31 ͔ and here we briefly review the essential nature of this phenomenon before specializing our discussion to the growth of symmetric dendrites. In Fig. 1 we illustrate changes in the crystallization morphology of PEO/PMMA blend films arising from a variation in the polymer composition. Crystallization was performed at 305 K and the clay concentration of the spin-casting solution was fixed at 5% of the mass of the blend. The PEO melting temperature T m depends on the polymer composition and T m values are indicated in the caption, along with the undercooling, ␦ T ϭ ( T m Ϫ T crys )/ T m . Clay particles can be seen at the center of the patterns shown in Fig. 1, confirming that the clay acts as a nucleating agent. Over a large range of PEO mass fraction ͑ 50–100 % of PEO by mass ͒ , we find circularly symmetric spherulites ͓ Fig. 1 ͑ b ͔͒ . This is the ‘‘normal’’ polymer crystallization morphology encountered under processing conditions ͓ 1,2,14 –16 ͔ . In the insert of Fig. 1 ͑ a ͒ , we show the late-stage spherulitic crystallization morphology where the spherulites impinge on each other and deform to form a do- main wall morphology similar in appearance to a Voronoi cell pattern ͓ 42 ͔ . The sidebranching of the spherulite ‘‘needles’’ becomes increasingly coarse with the increasing PMMA composition in this concentration regime ͓ 31 ͔ , but the spherulites tend to retain their nearly circular shape. At an almost 50/50 polymer blend ͑ PEO/PMMA ͒ mass ratio, we find a regime ͓ Fig. 1 ͑ b ͔͒ where the spherulite morphology changes into a seaweed dendrite morphology ͓ 19 ͔ . This morphology exhibits broad growing tips that split intermittently and one of the newly formed branches normally grows to predominate over the other. The dominant branch ͑ ‘‘alpha branch’’ ͒ then splits again and the process repeats itself. Tip splitting is the dominant feature of seaweed dendritic growth and this morphology is well known from recent modeling of nonequilibrium crystallization ͑ see below ͒ and has been confirmed in many experimental studies. Increasing the PMMA concentration further to near 30/70 leads to another dramatic change in the polymer crystallization morphology. In Fig. 1 ͑ c ͒ , we observe well-formed symmetric dendrites where the fourfold symmetry of equilibrium PEO crystallization asserts itself at a macroscopic scale ͓ 43 ͔ . The solution and the melt grown crystals have the same crystal structure ͑ square- shaped crystals ͒ under near-equilibrium conditions ͓ 43 ͔ . ͑ We have observed nearly square crystals in our films when we crystallize at 331 K near T m . Note the near registry of the sidebranches on each side of the growing dendrite arm and the uniformity of the ‘‘starlike’’ envelope curve describing the positions of the sidebranch tips of the dendrite. Symmetric dendritic polymer crystallization patterns have often been observed in polymer crystals grown on surfaces from polymer solutions ͓ 44 ͔ , but we are unaware of previous observations of SDC in melt blends. ͑ However, distorted spherulitic and randomly branched crystallization morphologies have been observed in melt blends ͓ 45 ͔ . ͒ At still higher concentrations of PMMA ͑ 20/80 ͒ , we observe another morphological transition from the symmetric dendritic crystallization to the highly branched, fractal morphology illustrated in Fig. 1 ͑ d ͒ ͓ 31 ͔ . This interesting transition is not well understood yet, but we do note that the high concentration of PMMA makes the film highly viscous and this could have an impact on the stability of the dendrite tips and the gross crystallization morphology. It is also notable that the growth should be diffusion-limited in this regime and the low concentration of PEO could also contribute to the noisy nature of the resulting crystal growth in this regime. In the following, we focus specifically on the dynamics of symmetric dendrite crystallization, corresponding to relative PEO/PMMA-mass concentration of 30/70 and 5% clay by mass, relative to the polymer. Figures 2 a –2 d show optical images false color of the growth of a ͑ PEO-rich ͒ symmetric dendrite over a sequence of times from 60 to 460 min. The film thickness is 160 nm and the dimensionless undercooling T ( T m T c )/ T m equals 0.01. Note the cusplike shape of the envelope curve describing the tip positions of the dendrite arms, a feature observed previously in symmetric dendrite growth at relatively high undercooling ͓ 46 ͔ . The sidebranches of the dendrite in Fig. 2 ͑ a ͒ grow nearly perpendicularly to the slender and nearly parabolic main branch of the dendrite. Our dendritic crystallization images were acquired at a rate of one picture every 5 min and in Fig. 2 ͑ b ͒ we show the increase in the tip position from the center of the dendrite. ͑ The clay seed at the dendrite center is the noticeable dark spot at the center of the dendrite. ͒ We observe that the tip position of the dendrite grows in an oscillatory manner about an average constant rate R o ϭ 0.171 ␮ m/min. The period P of the tip growth oscillations is of the order of 100 min. It is important to realize that the dendrite morphology changes in thinner films. Figure 2 ͑ c ͒ shows an example of dendritic growth in a 50-nm-thick film, where the crystallization conditions ͑ temperature, composition ͒ are the same as in Fig. 2 ͑ a ͒ . ͑ This morphological transition is discussed below. ͒ Notably, the dendrite in Fig. 2 ͑ c ͒ does not exhibit growth pulsations and has a more disordered appearance and its boundary envelope has a squarelike shape. The inset in Fig. 2 ͑ c ͒ shows an AFM image of the dendrite tip region, showing again a similarity to the form in the AFM and optical images. Despite differences in the large-scale crystallization morphology, the tip radius in Fig. 2 ͑ c ͒ is nearly the same as in ...

Context 10

... and general aspects of phase separation and crystallization in monotectic mixtures ͑ two liquids and one solid phase ͒ are discussed by Cahn ͓ 40 ͔ . Reflective optical images were obtained with an optical microscope ͑ OM ͒ using a Nikon optical microscope ͓ 32 ͔ with a digital Kodak MegaPlus, charge-coupled device cam- era attachment ͓ 32 ͔ . We follow the growth kinetics of the patterns using automated data acquisition with a resolution of 1024 ϫ 1024 pixels. All the atomic force microscopy ͑ AFM ͒ experiments were carried out in air by using a Dimension 3100 microscope from Digital Instruments operating in the Tapping modeTM ͓ 32 ͔ . In this mode, the cantilever is forced to oscillate at a frequency close to its resonance frequency with an adjustable amplitude. The tip, attached to the cantilever, was a pure silicon single crystal tip ͑ model TSEP ͒ with a radius of curvature of about 10 nm. The tip contacts briefly the film surface at each low position of the cantilever and the amplitude of the oscillation varies. ‘‘Height’’ images are obtained by using the feedback loop that keeps the amplitude at a constant value by translating vertically the sample with the pi- ezoelectric scanner: height measurements are deduced from those displacements. For the engagement we used a ratio A sp / A 0 ϭ 0.9, where A 0 is the free oscillation amplitude and A sp the set-point one ͓ 41 ͔ . The (512 ϫ 512 pixels) images have been obtained by using a (100 ϫ 100 ␮ m 2 ) piezoelec- tric scanner; the scanning frequency was 0.5 Hz and the mean value of the repulsive normal force was 0.1 nN. All the ‘‘height’’images have been filtered through the ‘‘Planefit’’ procedure ͓ 41 ͔ . We also used the AFM to measure the thickness of the film scratching the surface or masking a border of the wafer before spin casting the solution. The vertical resolution of AFM is 0.1 Å. In an earlier work, we investigated the real space structure of ͑ amorphous ͒ polymer-blend phase separation by forming nearly two-dimensional polymer films in which one of the polymer components segregates to both the solid substrate and the polymer-air boundary ͓ 30 ͔ . These ‘‘ultrathin’’ films were also restricted to film thickness range in which phase separation occurs within the plane of the blend film ͓ 30 ͔ . The difference in the surface tension between the blend components causes the film to buckle in response to phase separation within the film ͓ 30 ͔ . This buckling provides a good source of contrast in optical and AFM measurements, en- abling high-resolution measurements of the dynamics of the ordering process in real space. Here we extend this earlier work by incorporating a model crystallizable polymer ͑ PEO ͒ into the film to study dendritic growth. We mix PEO with an amorphous polymer ͑ PMMA ͒ , which is the component that segregates to the film boundaries. Clay particles are added as nucleating centers for the PEO crystallization. The clay particles are convenient because they induce centrosymmetric crystallization patterns. Crystallization can also be induced by scratching or piercing the film with a sharp implement so these particles are not essential for inducing crystallization. The crystallization morphology of PEO mixed with PMMA in a thin-film geometry can be ‘‘tuned’’ through spherulitic, seaweed, symmetric dendritic, and fractal dendritic patterns through the adjustment of the PMMA composition. These crystallization morphologies are described in a separate paper ͓ 31 ͔ and here we briefly review the essential nature of this phenomenon before specializing our discussion to the growth of symmetric dendrites. In Fig. 1 we illustrate changes in the crystallization morphology of PEO/PMMA blend films arising from a variation in the polymer composition. Crystallization was performed at 305 K and the clay concentration of the spin-casting solution was fixed at 5% of the mass of the blend. The PEO melting temperature T m depends on the polymer composition and T m values are indicated in the caption, along with the undercooling, ␦ T ϭ ( T m Ϫ T crys )/ T m . Clay particles can be seen at the center of the patterns shown in Fig. 1, confirming that the clay acts as a nucleating agent. Over a large range of PEO mass fraction ͑ 50–100 % of PEO by mass ͒ , we find circularly symmetric spherulites ͓ Fig. 1 ͑ b ͔͒ . This is the ‘‘normal’’ polymer crystallization morphology encountered under processing conditions ͓ 1,2,14 –16 ͔ . In the insert of Fig. 1 ͑ a ͒ , we show the late-stage spherulitic crystallization morphology where the spherulites impinge on each other and deform to form a do- main wall morphology similar in appearance to a Voronoi cell pattern ͓ 42 ͔ . The sidebranching of the spherulite ‘‘needles’’ becomes increasingly coarse with the increasing PMMA composition in this concentration regime ͓ 31 ͔ , but the spherulites tend to retain their nearly circular shape. At an almost 50/50 polymer blend ͑ PEO/PMMA ͒ mass ratio, we find a regime ͓ Fig. 1 ͑ b ͔͒ where the spherulite morphology changes into a seaweed dendrite morphology ͓ 19 ͔ . This morphology exhibits broad growing tips that split intermittently and one of the newly formed branches normally grows to predominate over the other. The dominant branch ͑ ‘‘alpha branch’’ ͒ then splits again and the process repeats itself. Tip splitting is the dominant feature of seaweed dendritic growth and this morphology is well known from recent modeling of nonequilibrium crystallization ͑ see below ͒ and has been confirmed in many experimental studies. Increasing the PMMA concentration further to near 30/70 leads to another dramatic change in the polymer crystallization morphology. In Fig. 1 ͑ c ͒ , we observe well-formed symmetric dendrites where the fourfold symmetry of equilibrium PEO crystallization asserts itself at a macroscopic scale ͓ 43 ͔ . The solution and the melt grown crystals have the same crystal structure ͑ square- shaped crystals ͒ under near-equilibrium conditions ͓ 43 ͔ . ͑ We have observed nearly square crystals in our films when we crystallize at 331 K near T m . Note the near registry of the sidebranches on each side of the growing dendrite arm and the uniformity of the ‘‘starlike’’ envelope curve describing the positions of the sidebranch tips of the dendrite. Symmetric dendritic polymer crystallization patterns have often been observed in polymer crystals grown on surfaces from polymer solutions ͓ 44 ͔ , but we are unaware of previous observations of SDC in melt blends. ͑ However, distorted spherulitic and randomly branched crystallization morphologies have been observed in melt blends ͓ 45 ͔ . ͒ At still higher concentrations of PMMA ͑ 20/80 ͒ , we observe another morphological transition from the symmetric dendritic crystallization to the highly branched, fractal morphology illustrated in Fig. 1 ͑ d ͒ ͓ 31 ͔ . This interesting transition is not well understood yet, but we do note that the high concentration of PMMA makes the film highly viscous and this could have an impact on the stability of the dendrite tips and the gross crystallization morphology. It is also notable that the growth should be diffusion-limited in this regime and the low concentration of PEO could also contribute to the noisy nature of the resulting crystal growth in this regime. In the following, we focus specifically on the dynamics of symmetric dendrite crystallization, corresponding to relative PEO/PMMA-mass concentration of 30/70 and 5% clay by mass, relative to the polymer. Figures 2 a –2 d show optical images false color of the growth of a ͑ PEO-rich ͒ symmetric dendrite over a sequence of times from 60 to 460 min. The film thickness is 160 nm and the dimensionless undercooling T ( T m T c )/ T m equals 0.01. Note the cusplike shape of the envelope curve describing the tip positions of the dendrite arms, a feature observed previously in symmetric dendrite growth at relatively high undercooling ͓ 46 ͔ . The sidebranches of the dendrite in Fig. 2 ͑ a ͒ grow nearly perpendicularly to the slender and nearly parabolic main branch of the dendrite. Our dendritic crystallization images were acquired at a rate of one picture every 5 min and in Fig. 2 ͑ b ͒ we show the increase in the tip position from the center of the dendrite. ͑ The clay seed at the dendrite center is the noticeable dark spot at the center of the dendrite. ͒ We observe that the tip position of the dendrite grows in an oscillatory manner about an average constant rate R o ϭ 0.171 ␮ m/min. The period P of the tip growth oscillations is of the order of 100 min. It is important to realize that the dendrite morphology changes in thinner films. Figure 2 ͑ c ͒ shows an example of dendritic growth in a 50-nm-thick film, where the crystallization conditions ͑ temperature, composition ͒ are the same as in Fig. 2 ͑ a ͒ . ͑ This morphological transition is discussed below. ͒ Notably, the dendrite in Fig. 2 ͑ c ͒ does not exhibit growth pulsations and has a more disordered appearance and its boundary envelope has a squarelike shape. The inset in Fig. 2 ͑ c ͒ shows an AFM image of the dendrite tip region, showing again a similarity to the form in the AFM and optical images. Despite differences in the large-scale crystallization morphology, the tip radius in Fig. 2 ͑ c ͒ is nearly the same as in Fig. 2 ͑ a ͒ , ( r Ϸ 1 ␮ m). The AFM data is discussed quantitatively below. We obtain some insight into these dendritic growth patterns by comparing to phase field simulations of two- dimensional SDC in a two-dimensional fluid mixture ͓ 47 ͔ . The simulation in Fig. 2 ͑ d ͒ corresponds to a Ni-Cu alloy ( ␾ Ni ϭ 0.59), where ␧ is taken to have a relatively large value, ␧ ϭ 0.05 and ␦ T is relatively large for metallurgical fluids, ␦ T ϭ 0.013. „ The new phase field calculation in Fig. 2 ͑ d ͒ is for ␧ ϭ 0.05, which is larger than the ␧ considered in previous work ͓ 47 ͔ ( ␧ ϭ 0.04), but otherwise the model parameters are identical to those specified in Ref. ...

Context 11

... on the dynamics of symmetric dendrite crystallization, corresponding to relative PEO/PMMA-mass concentration of 30/70 and 5% clay by mass, relative to the polymer. Figures 2 a –2 d show optical images false color of the growth of a ͑ PEO-rich ͒ symmetric dendrite over a sequence of times from 60 to 460 min. The film thickness is 160 nm and the dimensionless undercooling T ( T m T c )/ T m equals 0.01. Note the cusplike shape of the envelope curve describing the tip positions of the dendrite arms, a feature observed previously in symmetric dendrite growth at relatively high undercooling ͓ 46 ͔ . The sidebranches of the dendrite in Fig. 2 ͑ a ͒ grow nearly perpendicularly to the slender and nearly parabolic main branch of the dendrite. Our dendritic crystallization images were acquired at a rate of one picture every 5 min and in Fig. 2 ͑ b ͒ we show the increase in the tip position from the center of the dendrite. ͑ The clay seed at the dendrite center is the noticeable dark spot at the center of the dendrite. ͒ We observe that the tip position of the dendrite grows in an oscillatory manner about an average constant rate R o ϭ 0.171 ␮ m/min. The period P of the tip growth oscillations is of the order of 100 min. It is important to realize that the dendrite morphology changes in thinner films. Figure 2 ͑ c ͒ shows an example of dendritic growth in a 50-nm-thick film, where the crystallization conditions ͑ temperature, composition ͒ are the same as in Fig. 2 ͑ a ͒ . ͑ This morphological transition is discussed below. ͒ Notably, the dendrite in Fig. 2 ͑ c ͒ does not exhibit growth pulsations and has a more disordered appearance and its boundary envelope has a squarelike shape. The inset in Fig. 2 ͑ c ͒ shows an AFM image of the dendrite tip region, showing again a similarity to the form in the AFM and optical images. Despite differences in the large-scale crystallization morphology, the tip radius in Fig. 2 ͑ c ͒ is nearly the same as in Fig. 2 ͑ a ͒ , ( r Ϸ 1 ␮ m). The AFM data is discussed quantitatively below. We obtain some insight into these dendritic growth patterns by comparing to phase field simulations of two- dimensional SDC in a two-dimensional fluid mixture ͓ 47 ͔ . The simulation in Fig. 2 ͑ d ͒ corresponds to a Ni-Cu alloy ( ␾ Ni ϭ 0.59), where ␧ is taken to have a relatively large value, ␧ ϭ 0.05 and ␦ T is relatively large for metallurgical fluids, ␦ T ϭ 0.013. „ The new phase field calculation in Fig. 2 ͑ d ͒ is for ␧ ϭ 0.05, which is larger than the ␧ considered in previous work ͓ 47 ͔ ( ␧ ϭ 0.04), but otherwise the model parameters are identical to those specified in Ref. ͓ 47 ͔ . ... Com- parison of the simulation to our measurements is meant to be only qualitative. The main point is that growth pulsations are not observed in the two-dimensional phase field simulation, but we do find a reasonable resemblance between the ‘‘two- dimensional’’ polymer dendritic growth shown in Fig. 2 ͑ c ͒ and the simulated crystallization patterns ͓ Fig. 2 ͑ d ͔͒ . Apparently, no ␧ measurements have ever been made on high molecular weight polymers, and at present we are restricted to qualitative comparisons between the phase field model and our measurements. The oscillations in the tip radius in Fig. 2 ͑ b ͒ are quantified by subtracting the average dendrite tip radius R ( t ) ϭ R 0 t ͓ straight line in Fig. 2 ͑ b ͔͒ from R ( t ). Figure 3 ͑ a ͒ shows that the tip position fluctuation ␦ R ( t ) ϭ R ( t ) Ϫ R ( t ) is nearly sinusoidal; the solid curves correspond to a fit of ␦ R ( t ) to ␦ A R sin(2 ␲ t / P ), where ␦ A R is the oscillation amplitude and P is the pulsation period . We next compare ␦ R ( t ) to a measure of fluctuations in the width ␦ w ( t ) of the dendrite arm. To determine the dendrite sidebranch width w ( t ) we choose an arbitrary sidebranch ͓ denoted by arrow in Fig. 2 ͑ a ͔͒ and define w ( t ) as the orthogonal distance from the tip of the sidebranch to the center line of the main dendrite arm. The sidebranch width grows with an average rate ͑ equal to R o to within experimental uncertainty ͒ with oscillations ␦ w ( t ) about this average. Our determination of ␦ w ( t ) in Fig. 3 ͑ a ͒ shows that ␦ w ( t ) oscillates out of phase with ␦ R ( t ). Further, a ‘‘phase plot’’ of ␦ R ( t ) versus ␦ w ( t ) in Fig. 3 ͑ b ͒ reveals that the dendritic growth in Fig. 3 ͑ a ͒ is governed by a limit cycle with a phase angle ␣ difference of about 164° ͑ see caption of Fig. 3 ͒ . The dendritic growth in Fig. 2 ͑ a ͒ has a self-similar appearance and this suggests that it might be useful to determine the apparent mass-scaling dimension ͑ fractal dimension ͒ describing the symmetric polymer dendritic growth. The determination of fractal dimension d f was determined from image analysis based on the area-perimeter technique ͓ 48 ͔ . In Fig. 4 ͑ a ͒ , we show a plot of the polymer dendrite area A and perimeter p obtained by digitizing the optical image series corresponding to the dendrite growth shown in Fig. 1 ͑ a ͒ . We observe that a power law relation A ϳ p 1/ d f , can fit fairly well the data and we determine d f from the slope of log p versus log A . This gives an apparent fractal dimension d f ϭ 1.78 Ϯ 0.05, where the correlation coefficient for the power law fit is R 2 ϭ 0.98. Notably, we do not observe growth pulsations in Fig. 4 ͑ a ͒ so that A and p 1/1.78 must exhibit similar growth oscillations. The near linearity of the plot in Fig. 4 ͑ a ͒ confirms the impression of the near self- similarity of the polymer dendritic growth pattern. The temperature dependence of the rate of dendritic growth for a L ϭ 160 nm film is shown in Fig. 4 ͑ b ͒ as a function of undercooling, ⌬ T ϭ ( T m Ϫ T c ). Over the temperature range investigated, the rate of crystallization R 0 can be described by a power law, R 0 ϳ ( ⌬ T ) ␦ where ␦ Ϸ 2.53 Ϯ 0.02. The correlation coefficient for the power law fit is R 2 ϭ 0.99. A power scaling of R 0 with an effective exponent near 2.6 has been suggested to be a ‘‘universal’’ property of dendritic growth in small molecule liquids ͓ 49 ͔ . Since theory offers limited guidance about the factors governing the period P of dendrite growth pulsations, we explore the influence of some obvious system parameters under our control—undercooling, polymer composition ͑ supersaturation ͒ and film thickness, L . In Fig. 5 ͑ a ͒ we show ␦ R ( t ) for a range of undercooling ⌬ T ϭ ( T m Ϫ T c ) values in the range ͑ 288 –308 K ͒ . Apparently, P has no appreciable dependence on ⌬ T , but Fig. 5 ͑ b ͒ shows that ␦ A R increases nearly exponentially with ⌬ T . The correlation coefficient for the exponential fit is R 2 ϭ 0.99. A change in the relative polymer composition has a large influence on the pulsation rate, but this effect can cause a qualitative change in the crystallization morphology ͓ 31 ͔ so we restrict ourselves to a composition range where the SDC is observed. A decrease of the PMMA concentration causes a decrease in p and an increase in ␦ A R for a fixed ⌬ T ϭ 35 K, ͑ e.g., P Ϸ 105 min and ␦ A R Ϸ 15 ␮ m for a 35/65 blend while P Ϸ 180 min and ␦ A R Ϸ 9 ␮ m for a 30/70 blend ͒ . We next explore the change in the growth pattern dynamics and morphology associated with reducing the film’s thickness. Figure 6 shows that P first increases sharply with decreasing film thickness L , but then drops precipitously to zero below a critical film thickness, L c Ϸ 80 nm. The SDC is similar to Fig. 1 ͑ a ͒ for L Ͼ L c , but we observe a different morphology for L Ͻ L c ͑ see Fig. 2 ͑ c ͒ and inset to Fig. 6 ͒ . Thus, we have direct evidence that the morphological transition is accompanied by a change in the dynamics of the dendrite tip. The lack of pulsations in the ‘‘two-dimensional’’ blend film dendrites is also reflected in the extent of correlation in the position of the sidebranches on each side of the primary growing parabolic dendrite arms ͓ see Fig. 2 ͑ c ͔͒ . The registry of sidebranches and the cusplike envelope curve describing the positions of the sidebranch tips in the symmetric dendrite shown in Fig. 2 ͑ a ͒ are contrasted with the ‘‘two- dimensional’’ dendrite ( L Ͻ L c ) shown in Fig. 2 ͑ c ͒ . There is little correlation in the sidebranch positions on either side of this dendrite. This enhanced regularity of structure in the pulsing dendrite is reminiscent of the regular sidebranching found in the growth of dendritic growth subjected to periodic external perturbations ͓ 50,51 ͔ . We, therefore, suggest that the oscillatory tip mode imparts regularity to the growing dendrite. Finally, we should mention in this section that oscillatory growth front modes have recently been reported in the spherulite and seaweed crystallization morphologies for other materials ͑ Fig. 1; see Discussion ͒ so that the presence of hydrodynamic modes in propagating crystallization fronts appears to be a general, but nonuniversal, phenomenon. However, the study of the dynamics of these other nonequilibrium crystallization morphologies will require the devel- opment of specialized measurement techniques for each morphology ͑ the splitting of the dendrite tip creates some ambiguity in defining the precise location of the crystallization front ͒ , so in the present paper, we confine our attention to symmetric dendrite growth. The singular nature of the shift in P with L provides an important clue into the nature of the dendritic growth pulsa- tions. At first, we anticipated that P would correspond to a diffusion-controlled depletion time that would scale quadrati- cally with film thickness. This expectation would lead to a decrease of P with film thickness; an effect opposite to our measurements. We then realized that the L dependence of P is similar to the finite-size dependence of pulsations observed in oscillatory chemical reactions. This comparison is natural because Belousov-Zhabotinsky ͑ BZ ͒ reactions also exhibit pattern formation with propagating wave fronts. The oscillation period of the BZ ...

Context 12

... in Fig. 2 ͑ d ͒ is for ␧ ϭ 0.05, which is larger than the ␧ considered in previous work ͓ 47 ͔ ( ␧ ϭ 0.04), but otherwise the model parameters are identical to those specified in Ref. ͓ 47 ͔ . ... Com- parison of the simulation to our measurements is meant to be only qualitative. The main point is that growth pulsations are not observed in the two-dimensional phase field simulation, but we do find a reasonable resemblance between the ‘‘two- dimensional’’ polymer dendritic growth shown in Fig. 2 ͑ c ͒ and the simulated crystallization patterns ͓ Fig. 2 ͑ d ͔͒ . Apparently, no ␧ measurements have ever been made on high molecular weight polymers, and at present we are restricted to qualitative comparisons between the phase field model and our measurements. The oscillations in the tip radius in Fig. 2 ͑ b ͒ are quantified by subtracting the average dendrite tip radius R ( t ) ϭ R 0 t ͓ straight line in Fig. 2 ͑ b ͔͒ from R ( t ). Figure 3 ͑ a ͒ shows that the tip position fluctuation ␦ R ( t ) ϭ R ( t ) Ϫ R ( t ) is nearly sinusoidal; the solid curves correspond to a fit of ␦ R ( t ) to ␦ A R sin(2 ␲ t / P ), where ␦ A R is the oscillation amplitude and P is the pulsation period . We next compare ␦ R ( t ) to a measure of fluctuations in the width ␦ w ( t ) of the dendrite arm. To determine the dendrite sidebranch width w ( t ) we choose an arbitrary sidebranch ͓ denoted by arrow in Fig. 2 ͑ a ͔͒ and define w ( t ) as the orthogonal distance from the tip of the sidebranch to the center line of the main dendrite arm. The sidebranch width grows with an average rate ͑ equal to R o to within experimental uncertainty ͒ with oscillations ␦ w ( t ) about this average. Our determination of ␦ w ( t ) in Fig. 3 ͑ a ͒ shows that ␦ w ( t ) oscillates out of phase with ␦ R ( t ). Further, a ‘‘phase plot’’ of ␦ R ( t ) versus ␦ w ( t ) in Fig. 3 ͑ b ͒ reveals that the dendritic growth in Fig. 3 ͑ a ͒ is governed by a limit cycle with a phase angle ␣ difference of about 164° ͑ see caption of Fig. 3 ͒ . The dendritic growth in Fig. 2 ͑ a ͒ has a self-similar appearance and this suggests that it might be useful to determine the apparent mass-scaling dimension ͑ fractal dimension ͒ describing the symmetric polymer dendritic growth. The determination of fractal dimension d f was determined from image analysis based on the area-perimeter technique ͓ 48 ͔ . In Fig. 4 ͑ a ͒ , we show a plot of the polymer dendrite area A and perimeter p obtained by digitizing the optical image series corresponding to the dendrite growth shown in Fig. 1 ͑ a ͒ . We observe that a power law relation A ϳ p 1/ d f , can fit fairly well the data and we determine d f from the slope of log p versus log A . This gives an apparent fractal dimension d f ϭ 1.78 Ϯ 0.05, where the correlation coefficient for the power law fit is R 2 ϭ 0.98. Notably, we do not observe growth pulsations in Fig. 4 ͑ a ͒ so that A and p 1/1.78 must exhibit similar growth oscillations. The near linearity of the plot in Fig. 4 ͑ a ͒ confirms the impression of the near self- similarity of the polymer dendritic growth pattern. The temperature dependence of the rate of dendritic growth for a L ϭ 160 nm film is shown in Fig. 4 ͑ b ͒ as a function of undercooling, ⌬ T ϭ ( T m Ϫ T c ). Over the temperature range investigated, the rate of crystallization R 0 can be described by a power law, R 0 ϳ ( ⌬ T ) ␦ where ␦ Ϸ 2.53 Ϯ 0.02. The correlation coefficient for the power law fit is R 2 ϭ 0.99. A power scaling of R 0 with an effective exponent near 2.6 has been suggested to be a ‘‘universal’’ property of dendritic growth in small molecule liquids ͓ 49 ͔ . Since theory offers limited guidance about the factors governing the period P of dendrite growth pulsations, we explore the influence of some obvious system parameters under our control—undercooling, polymer composition ͑ supersaturation ͒ and film thickness, L . In Fig. 5 ͑ a ͒ we show ␦ R ( t ) for a range of undercooling ⌬ T ϭ ( T m Ϫ T c ) values in the range ͑ 288 –308 K ͒ . Apparently, P has no appreciable dependence on ⌬ T , but Fig. 5 ͑ b ͒ shows that ␦ A R increases nearly exponentially with ⌬ T . The correlation coefficient for the exponential fit is R 2 ϭ 0.99. A change in the relative polymer composition has a large influence on the pulsation rate, but this effect can cause a qualitative change in the crystallization morphology ͓ 31 ͔ so we restrict ourselves to a composition range where the SDC is observed. A decrease of the PMMA concentration causes a decrease in p and an increase in ␦ A R for a fixed ⌬ T ϭ 35 K, ͑ e.g., P Ϸ 105 min and ␦ A R Ϸ 15 ␮ m for a 35/65 blend while P Ϸ 180 min and ␦ A R Ϸ 9 ␮ m for a 30/70 blend ͒ . We next explore the change in the growth pattern dynamics and morphology associated with reducing the film’s thickness. Figure 6 shows that P first increases sharply with decreasing film thickness L , but then drops precipitously to zero below a critical film thickness, L c Ϸ 80 nm. The SDC is similar to Fig. 1 ͑ a ͒ for L Ͼ L c , but we observe a different morphology for L Ͻ L c ͑ see Fig. 2 ͑ c ͒ and inset to Fig. 6 ͒ . Thus, we have direct evidence that the morphological transition is accompanied by a change in the dynamics of the dendrite tip. The lack of pulsations in the ‘‘two-dimensional’’ blend film dendrites is also reflected in the extent of correlation in the position of the sidebranches on each side of the primary growing parabolic dendrite arms ͓ see Fig. 2 ͑ c ͔͒ . The registry of sidebranches and the cusplike envelope curve describing the positions of the sidebranch tips in the symmetric dendrite shown in Fig. 2 ͑ a ͒ are contrasted with the ‘‘two- dimensional’’ dendrite ( L Ͻ L c ) shown in Fig. 2 ͑ c ͒ . There is little correlation in the sidebranch positions on either side of this dendrite. This enhanced regularity of structure in the pulsing dendrite is reminiscent of the regular sidebranching found in the growth of dendritic growth subjected to periodic external perturbations ͓ 50,51 ͔ . We, therefore, suggest that the oscillatory tip mode imparts regularity to the growing dendrite. Finally, we should mention in this section that oscillatory growth front modes have recently been reported in the spherulite and seaweed crystallization morphologies for other materials ͑ Fig. 1; see Discussion ͒ so that the presence of hydrodynamic modes in propagating crystallization fronts appears to be a general, but nonuniversal, phenomenon. However, the study of the dynamics of these other nonequilibrium crystallization morphologies will require the devel- opment of specialized measurement techniques for each morphology ͑ the splitting of the dendrite tip creates some ambiguity in defining the precise location of the crystallization front ͒ , so in the present paper, we confine our attention to symmetric dendrite growth. The singular nature of the shift in P with L provides an important clue into the nature of the dendritic growth pulsa- tions. At first, we anticipated that P would correspond to a diffusion-controlled depletion time that would scale quadrati- cally with film thickness. This expectation would lead to a decrease of P with film thickness; an effect opposite to our measurements. We then realized that the L dependence of P is similar to the finite-size dependence of pulsations observed in oscillatory chemical reactions. This comparison is natural because Belousov-Zhabotinsky ͑ BZ ͒ reactions also exhibit pattern formation with propagating wave fronts. The oscillation period of the BZ reaction occurring in ion- exchange beads ͓ 52 ͔ ͑ which causes the color of the beads to flicker ͒ likewise increases strongly with decreasing bead radius and the oscillations cease when the bead size became smaller than a critical radius ͑ 0.2 mm ͒ . In Fig. 6, we compare our measurements of P to the functional form suggested by the studies of finite-size effects on the BZ reaction ͑ bead radius is replaced by polymer film thickness ͒ ͓ 52 ͔ . This leads to the relation, P ϭ P ρ /(1 Ϫ L / L c ) for L Ͼ L c , P ρ ϭ 90 min and P ϭ 0 for L Ͻ L c . The correlation coefficient for the data point fit is R 2 ϭ 0.99. The finite-size dependence of the oscillation period in the BZ reaction was attributed in Ref. ͓ 51 ͔ to a change in the reaction rate due to the inactive nature of the reaction at the bead surface, leading to a correction of the reaction rate involving the surface-volume ratio. In our own measurements, the boundaries of the blend film are enriched in PMMA so that a similar finite-size effect on the pulsation period is plausible. The viewpoint of a supercooled liquid as a variety of the excitable medium and crystallization as a variety of reaction- diffusion wave propagation also gives insight into the influence of the clay particles on the crystallization morphology. At low concentrations, the clay particles mainly serve as centers of the dendritic growth and similar dendritic patterns can be obtained by punching or scratching the film without clay. The ‘‘catalyst’’ particles play a similar role as an excitation source for BZ reactions in solutions loaded with ferroin- loaded resin beads ͓ 53 ͔ . As is well known, the BZ reaction in a fluid layer gives rise to symmetric ‘‘target’’ chemical waves at low bead concentrations, but these patterns break up into rotating spiral patterns at higher bead concentrations due to the interference between the chemical waves ͓ 53–55 ͔ . Our proposed analogy between nonequilibrium crystallization and autocatalytic chemical reactions would lead us to expect a similar symmetry breaking phenomenon in dendritic ...