Observation of the "burst-like growth" regime for the 4 He crystals nucleated in a metastable liquid
aa r X i v : . [ c ond - m a t . o t h e r] N ov Observation of the ”burst-like growth” regime for the He crystals nucleated in ametastable liquid.
V. L. Tsymbalenko ∗ NRC Kurchatov Institute, 123182 sq.Kurchatov 1Institute for Physical Problems RAS, 119334 Kosigin st.2Moscow, Russia
The ”burst-like growth” regime is observed for the He crystals with the growth defects. Theobservation has confirmed the hypothesis for the same physical mechanisms responsible for thetransition of the crystalline facets to the state of abnormally fast growth at high and low tempera-tures. The relaxation process of the kinetic growth coefficient is found to be similar to the relaxationof the elastic modules of the crystal at the end of the fast growth stage. The kinetic growth coef-ficients are determined at the stages of fast and slow growth. The crossover from the fast to slowkinetics of crystal facet growth is found to be drastic.
PACS numbers: 68.08.-p, 67.80.-s, 68.35.Rh
INTRODUCTION
In 1996 the amazing growth of helium crystals is dis-covered in two different series of experiments. In thefirst series the phenomenon is observed in the quasi pe-riodic growth of the perfect crystalline c-facet within therange 2-250 mK. This is the so-called ”burst-like growth”[1, 2]. Under constant liquid flow into the container thepressure increases since the crystal facet does not growdue to lack of growth sources. However, as the super-saturation reaches the order of tenths of millibar, thecrystal facet starts to grow suddenly. The displacementof the facet is observed by the optical methods. The fastgrowth results in a drastic pressure drop. Then the facetbecomes immobile, the pressure enhances and the cyclerepeats.In the second series at temperatures 0.4-0.75K the He crystal nucleates in metastable superfluid liquid [3].Within this temperature range the growth kinetics is gov-erned by the most slowly growing sections of the surface,i.e., c- and a-facets. The crystal grows as a hexagonalprism. After nucleation, the crystal grows slowly. Thegrowth is accompanied with the slow pressure drop inthe container. Then, the kinetic growth coefficient of allfacets increases drastically by 2-3 orders of the magni-tude. As a result, the whole crystal grows rapidly within ∼ µs , Its growth is accompanied by the drastic pres-sure drop. The shape and size of the crystal at this stageis recorded by video filming.In the both series of experiments the phenomenonis observed when certain temperature-dependent magni-tude of supersaturation p b is exceeded. The phase di-agrams, showing the regions of normal and abnormalgrowth for the different temperature ranges, do not con-tradict each other. We observe the similar stochasticnature for the onset of the fast growth stage and for theeffect of a small He impurity on the boundary supersat-uration p b . This similarity makes it possible to proposethat the crystal growth acceleration under different ex- perimental conditions leads to the same mechanism. Ifthis hypothesis is correct, the burst-like growth modeshould be observed at the temperatures above 250 mK.A review of experimental techniques, results obtained forphase diagrams p b ( T ) of the burst like growth, statisticsof its nucleation, facets growth rates, elastic modulus re-laxation of a crystal after the end of rapid growth, etc.is given in Ref. [4].To initiate the ”burst-like growth” mode for the equi-librium crystals at T=0.4-0.75 K, the supersaturationshould be produced exceeding the p b boundary super-saturation equal to 2-10mbar at these temperatures. Anattempt to create such conditions in the dynamic mannerhas failed when the crystal is encircled with the super-fluid helium, see Ref. [4], Chapter 4.8. The next attemptis made for the crystals in the normal state. The point isthat, in contrast to perfect crystals in experiments [1, 2],the facets of such crystals have growth defects [5]. Asa result, when liquid is pumped into the container, thecrystal grows absorbing most of the inflowing liquid. Inaddition, the high capillary impedance, the large volumeof the container 4–10 cm , and the limited capabilities ofthe external pressure system do not allow us to producea high liquid flow sufficient to compensate for the crystalgrowth. As a result, the supersaturation required is notachieved.This paper presents the results of measurements with atechnique that overcomes partially the limitations of theprevious experiments. The internal volume of the con-tainer is reduced by a factor ∼
50 and the impedance ofthe inlet capillary is reduced by one order of magnitude.
EXPERIMENTAL TECHNIQUE AND THERESULTS
The crystals are grown in a container according to themethod described in Ref. [4], Chapter 2. The internal vol-ume of the container is 80 mm . A tungsten needle for K , s / m ∆ p0, mbar FIG. 1: Dependence of the average kinetic growth coefficienton the initial supersaturation ∆ p at T = 0.74K. Open circlescorrespond to the crystals with anomalously fast growth. Thesolid circles are crystals with normal slow kinetics of the facetgrowth due to defects. The half-filled circle shows the magni-tude of the kinetic growth rate calculated from the envelope∆ p ( t ). See Fig.2, upper graph, lower curve. the crystal nucleation in a metastable liquid He is placedat the center of the container. One of the container wallsrepresents membrane of capacitive sensor with the timelag 160 µs . The main difference between this techniqueand the previous one is that the nucleation of a crystaland its next growth take place under continuous liquidflow into the container. The experiments are carried outat two temperatures 0.49K and 0.74K. The upper limit ofsupersaturation is limited by spontaneous crystal nucle-ation on the inner wall of the container. In these exper-iments, the supersaturation magnitudes are within therange 0.1 – 5 mbar. The kinetic growth coefficient K isdetermined by the expression V surf = K ∆ ρρρ ′ ∆ p (1)where V surf is the surface growth rate, ρ and ρ ′ are thedensities of liquid and solid helium, ∆ ρ = ρ ′ − ρ >
0. Theoverpressure ∆ p is measured from the phase equilibriumpressure.The supersaturation ∆ p at crystal nucleation deter-mines its growth mode. Figure 1 shows the magnitudes ofthe kinetic growth coefficient for the series of crystals at0.74 K. It can be seen that the crystals can be separatedinto two groups. The crystals that start growing withthe supersaturation less than ∼ K = 0 . ± . ∼ . ± . p b lies within the range 2.5-3.5 mbar.
50 60 70 80 90 1000.00.20.40.60.8
50 60 70 80 90 1000.000.050.100.150.20 ∆ p , m ba r t, ms ∆ p , m ba r t, ms FIG. 2: Pressure changes in the container during crystalgrowth. The upper graph refers to a temperature of 0.74K,the lower one - 0.49K. The curves with jumps at t = 0 in bothplots show the pressure change with the growth of a partic-ular crystal. The smooth curve, shifted lower by the arrow,is the result of averaging over a series of measurements withthe same starting conditions. The inset, on an enlarged scale,shows burst-like growth like pressure surges, Refs. [1, 2].
This magnitude is a half of the boundary supersatura-tion separating the regions of normal and abnormal crys-tal growth studied earlier, see Fig.11 in Ref. [4]. ForT = 0.49K, this boundary lies below 1 mbar which isalso much lower than the magnitude p b ≈ ∼
10 ms. The evaluation for a ratio of theamplitude of the first oscillations period to the start-ing pressure gives the magnitude K = 6-7 s/m at thistemperature. The magnitudes of the kinetic growth co-efficient and their temperature dependence are consistentwith the results obtained earlier [4].The liquid flow into the container forces the crystalto grow continuously. If the kinetic growth coefficient ofcrystal facets is small, the growth occurs at significant su-persaturation ∆ p . Provided that the facets switch ontothe ”burst-like growth” mode with the kinetic growthcoefficient K larger by the orders of the magnitude, thesupersaturation proves to be very small. The pressurerecords show both of these processes. The return to theslow kinetics of crystal surface growth starts after com-pleting of the fast growth stage. The supersaturationincreases, passes through a maximum in the region of ∼ ∼ p ( t ) curves obtained by averaging over aseries of measurements. The pressure records for thecrystals nucleated under same conditions are aligned atthe onset of growth, summed up, and normalized. Suchtreatment averages the quasiperiodic pressure jumps.The lower curves in Fig.2 correspond to the envelopeof the ∆ p ( t ) dependences. Figure 3 shows the depen-dences of the kinetic growth coefficient K calculated bythe method in Ref. [6]. At first, the relaxation kinet-ics is seen, corresponding to the rise in pressure at theaveraged curves in Fig.2. The time constant at a tem-perature of 0.74K is 5 ms and at T = 0.49K is 7 ms.After ∼ K = 0 . ± .
006 s/m, see the half-filled circle inFig.1. The cooling to 0.49K increases growth coefficient K almost twice as large to K = 0 . ± .
007 s/m.The records in Fig.2 show the pressure jumps in theinterval 30-250 ms. The sharp pressure drop evidencesfor the fast crystal growth. The subsequent increase insupersaturation means a return to slow kinetics. Fromthe record ∆ p ( t ) in Fig.2 one can see that the pres-sure jumps appear at the supersaturations lying withinrange 0.3-0.8 mbar at temperature 0.74 K. The decreasein temperature shifts this interval down to magnitudes0.05-0.2 mbar. The temperature affects the frequency ofpressure jumps, as can be seen from Fig.2 in the insets.For T = 0.49K, the pressure jumps are formed more fre-quently than at T = 0.74K. The shape of the pressurejumps is asymmetric. After the onset of the burst-likegrowth state at T=0.74K, the pressure drop occurs in ∼ µs . This time is determined by the kinetic growthcoefficient and corresponds to the fast growth kinetics af-ter nucleating the crystal. The supersaturation returnsto its original magnitude in 4-5 ms. For temperature0.49K, the pressure drop time is 200 − µs . Note in K , s / m t, ms FIG. 3: Changes in the kinetic growth coefficient over time.The solid curve is calculated from the envelope of a series ofrecords at T = 0.74K. Dashed line - along the series at T =0.49K. this case that the time is not determined with the kineticgrowth coefficient but with the frequency of soft modesof pressure oscillations during crystal growth [4], Chap-ter 2.5. The supersaturation recovery time decreases to0.6-0.8 ms. After ∼
250 ms, the supersaturation decreasesand the burst-like growth process stops.
DISCUSSION
The experiments have shown that the burst-like growthregime, observed previously for the perfect crystals, isalso realized for the crystals with growth defects. Forhigh temperatures 0.4-0.75K and for low temperatures of2-250 mK, the effect of anomalously fast crystal growthmanifests itself in the same way. This is a strong argu-ment for the physical identity of the both effects. Thoughthis conclusion does not clarify the physical mechanismresponsible for the effect, nevertheless it allows us to elim-inate many possible explanations. For example, since theeffect is observed for the perfect crystal facet [1, 2], allthe possible explanations associated with the presence ofgrowth defects should be disregarded.The difference between the present experimental setupand the previous one is the container volume. The vol-ume is reduced by ∼ ∼ p b is related tothe crystal volume.The magnitudes of the kinetic growth coefficient at thefast growth stage, calculated from the pressure drop afterthe crystal nucleation, agree well with the magnitudesmeasured before. That is, the fast growth of the crystals,large and small in some meaning, is similar.The previous experiments performed within range0.48-0.68K have only shown the general features for re-turning the crystal facet growth kinetics to the normalregime [6]. It is shown that the significant relaxationtakes place for time ∼
20 ms after ending the fast growthstage. Next, the slow relaxation is observed to the nor-mal magnitudes of the kinetic growth coefficient. Thedetails of this process have not been clarified due to ex-perimental limitations indicated in the Introduction.The reduction of the kinetic growth rate within ∼
30 msafter the fast growth stage demonstrates the relaxationof the burst-like growth state to the normal state of lowgrowth kinetics. Note that for the same time, the re-laxation of the real and imaginary parts of the crystalelastic modulus takes place, see Ref. [4], Chapter 4.7.2.The relaxation time of this process changes insignificantlywithin the range 0.4-0.75K and takes 3-4 ms. Unfortu-nately, we do not know either this coincidence is occa-sional or this is two aspects of the same process since noanswer can be obtained from the data available.The magnitudes for the kinetic growth coefficient ofthe crystals grown in the normal state are one order ofmagnitude lower than the growth coefficients at the fastgrowth stage of fast growth, see Fig.1. The growth coef-ficient measurements performed by a number of authorsat about 0.75 K have a wide dispersion of these mag-nitudes from 4 ∗ − to 0.02 s/m, see Fig.4 [4]. Thedispersion is not surprising since the crystal facets growdue to growth defects in this stage. The defect structuredepends essentially on the conditions of crystal growth,e.g. annealing, etc. The conditions are different in var-ious experiments. The effect of different concentrationsof growth defects on the kinetics is clearly seen in thedifferent growth rates of the equivalent a-facets in theprocess of free crystal growth [5]. It is possible thatthe high growth rates of ”small” crystals are associated with their non-ideal structure. Note that the stationarymagnitudes of the kinetic growth coefficient, calculatedfrom the pressure envelope, are one order of the magni-tude smaller than those for the normal crystal growth inthese experiments, see Fig.3. The magnitudes are close tothose measured previously in the works of other authorsTo summarize, we have succeeded to reproduce theburst-like growth mode in the crystals containing thegrowth defects. This means that the physical mecha-nisms responsible for the crossover of the facets to thestate of abnormally fast growth are identical. The relax-ation process for the kinetic growth coefficient is found tobe similar to the relaxation of the crystal elastic modulesat the end of the fast growth stage. The kinetic growthfactors at the fast and slow growth stages have been de-termined. The crossover from the fast to slow kinetics isfound to be drastic as well. ACKNOWLEDGMENTS
The author is grateful to V. V. Dmitriev for the pos-sibility of performing these experiments at Kapitza In-stitute for Physical Problems RAS. The author is alsograteful to V. V. Zavyalov for supporting this work,S.N.Burmistrov for helpful comments and V. S. Kruglovfor interest to the work. ∗ [email protected][1] A. V. Babkin, P. J. Hakonen, A. Ya. Parshin, J. S. Pent-tila, J. P. Ruutu, J. P. Saramaki, G. Tvalashvili, Phys.Rev.Lett. , , 4187 (1996)[2] A. V. Babkin, P. J. Hakonen, A. Ya. Parshin, J. P. Ruutu,G. Tvalashvili, J.Low Temp.Phys. , , 117 (1998)[3] V. L. Tsymbalenko, Phys.Lett.A , , 177 (1996)[4] V. L. Tsymbalenko, Physics-Uspekhi , , 1059 (2015)[5] V. L. Tsymbalenko, Ukr.Low Temp.Phys. , , 120 (1995)[6] V. L. Tsymbalenko, JETP ,99