A Simplified Scaling Law of Cell-Dendrite Transition in...
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Review Article A Simplified Scaling Law of Cell-Dendrite Transition in Directional Solidification Yaochan Zhu , 1,2 Hua Qiu, 1 Zhijun Wang, 2 and Eckart Schnack 3 1 Xi’an Aeronautical Polytechnic Institute, Xi’an 710089, China 2 State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an, China 3 Karlsruhe Institute of Technology, Karlsruhe 76131, Germany Correspondence should be addressed to Yaochan Zhu; [email protected] Received 11 February 2019; Revised 11 May 2019; Accepted 21 May 2019; Published 2 June 2019 Academic Editor: Kiyokazu Yasuda Copyright © 2019 Yaochan Zhu et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. To describe the cell-dendrite transition (CDT) during directional solidification, a new simplified scaling law is proposed and verified by quantitative phase field simulations. is scaling law bears clear physical foundation with consideration of the overall effects of primary spacing, pulling velocity, and thermal gradient on the onset of sidebranches. e analysis results show that the exponent parameters in this simplified scaling law vary within different systems, which mediates the discrepancy of exponent parameters in previous experiments. e scaling law also presents an explanation for the destabilizing mechanism of thermal gradient in sidebranching dynamics. 1. Introduction As one of the typical problems of pattern formation, solid- ification microstructures are interesting for both scientific and technologic reasons [1–3]. ough remarkable progress in the study of microstructure evolution during solidifi- cation has been made, the mechanism of sidebranching dynamics is still unsatisfactorily understood [4–6]. During directional solidification, the onset of sidebranches means the occurrence of cell-dendrite transition (CDT). Over past few years, the identification of CDT has been widely performed experimentally [7–11], which showed that cell and dendrite coexist over a large range of control parameters. Given pulling velocity and thermal gradient, small primary spacing corresponds to cellular morphology in CDT region, while large primary spacing corresponds to dendrite. Accordingly, CDT significantly depends on the primary cellular/dendritic spacing, and a critical primary spacing 1 corresponding to CDT should exist. In terms of the correlation between 1 and control parameters, researchers have put much effort all along on finding the criteria of CDT by considering the spacing, pulling velocity , and thermal gradient . Based on abundant experiments [7–10], an empirical scaling law with the form of 1 ∞ has been summarized to describe the critical primary spacing 1 in which and are exponent parameters. Although the form of scaling law has been proposed, some controversies still exist. On one hand, the values of the exponent and are inconsistent in different experiments. = −1/2 and = −1/8 were found in Gerogelin et al.’s experiments [7] and = = −1/3 in Trivedi et al.’s experiments [8, 9]. On the other hand, this empirical scaling law was based on the data fitting without further physical foundation. Researchers have tried to expound the intrinsic physical foundation of the scaling law. In Trivedi et al.’s work [8], the critical primary spacing 1 was given by the geometrical meaning of three characteristic lengths: solutal diffusion length , thermal length , and capillary length 0 . Gerogelin et al. [7] also presented a self- similar asymptotic regime about / , 1 / 0 , and / 0 based on their experimental data. However, these analyses only focused on the assembly of characteristic lengths, not referring to the sidebranching dynamics, which characterizes the CDT. e exponent parameters selection and the physical foundation in the scaling law of CDT are still unclear and need further exploration. e crossover of CDT is usually defined by the occurring of sidebranches. Accordingly, sidebranching should be one Hindawi Advances in Condensed Matter Physics Volume 2019, Article ID 8767640, 8 pages https://doi.org/10.1155/2019/8767640
Transcript of A Simplified Scaling Law of Cell-Dendrite Transition in...
Review ArticleA Simplified Scaling Law of Cell-Dendrite Transition inDirectional Solidification
Yaochan Zhu 12 Hua Qiu1 ZhijunWang2 and Eckart Schnack3
1Xirsquoan Aeronautical Polytechnic Institute Xirsquoan 710089 China2State Key Laboratory of Solidification Processing Northwestern Polytechnical University Xirsquoan China3Karlsruhe Institute of Technology Karlsruhe 76131 Germany
Correspondence should be addressed to Yaochan Zhu zhuyaocan1978hotmailcom
Received 11 February 2019 Revised 11 May 2019 Accepted 21 May 2019 Published 2 June 2019
Academic Editor Kiyokazu Yasuda
Copyright copy 2019 Yaochan Zhu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
To describe the cell-dendrite transition (CDT) during directional solidification a new simplified scaling law is proposed and verifiedby quantitative phase field simulations This scaling law bears clear physical foundation with consideration of the overall effects ofprimary spacing pulling velocity and thermal gradient on the onset of sidebranches The analysis results show that the exponentparameters in this simplified scaling law vary within different systems which mediates the discrepancy of exponent parametersin previous experiments The scaling law also presents an explanation for the destabilizing mechanism of thermal gradient insidebranching dynamics
1 Introduction
As one of the typical problems of pattern formation solid-ification microstructures are interesting for both scientificand technologic reasons [1ndash3] Though remarkable progressin the study of microstructure evolution during solidifi-cation has been made the mechanism of sidebranchingdynamics is still unsatisfactorily understood [4ndash6] Duringdirectional solidification the onset of sidebranchesmeans theoccurrence of cell-dendrite transition (CDT) Over past fewyears the identification of CDT has been widely performedexperimentally [7ndash11] which showed that cell and dendritecoexist over a large range of control parameters Givenpulling velocity and thermal gradient small primary spacingcorresponds to cellular morphology in CDT region whilelarge primary spacing corresponds to dendrite AccordinglyCDT significantly depends on the primary cellulardendriticspacing and a critical primary spacing1205821119862119863 corresponding toCDT should exist In terms of the correlation between 1205821119862119863and control parameters researchers have put much effortall along on finding the criteria of CDT by considering thespacing pulling velocity119881 and thermal gradient119866 Based onabundant experiments [7ndash10] an empirical scaling law withthe form of 1205821119862119863infin119881120572119866120573 has been summarized to describe
the critical primary spacing 1205821119862119863 in which 120572 and 120573 areexponent parameters
Although the form of scaling law has been proposedsome controversies still exist On one hand the values of theexponent 120572 and 120573 are inconsistent in different experiments120572 = minus12 and 120573 = minus18 were found in Gerogelin etalrsquos experiments [7] and 120572 = 120573 = minus13 in Trivedi etalrsquos experiments [8 9] On the other hand this empiricalscaling law was based on the data fitting without furtherphysical foundation Researchers have tried to expound theintrinsic physical foundation of the scaling law In Trivediet alrsquos work [8] the critical primary spacing 1205821119862119863 wasgiven by the geometrical meaning of three characteristiclengths solutal diffusion length 119897119863 thermal length 119897119879 andcapillary length 1198890 Gerogelin et al [7] also presented a self-similar asymptotic regime about 119897119879119897119863 12058211198621198631198890 and 1198971198791198890based on their experimental data However these analysesonly focused on the assembly of characteristic lengths notreferring to the sidebranching dynamics which characterizesthe CDTThe exponent parameters selection and the physicalfoundation in the scaling law of CDT are still unclear andneed further exploration
The crossover of CDT is usually defined by the occurringof sidebranches Accordingly sidebranching should be one
HindawiAdvances in Condensed Matter PhysicsVolume 2019 Article ID 8767640 8 pageshttpsdoiorg10115520198767640
2 Advances in Condensed Matter Physics
of the typical characteristics in CDT and the connectionsbetween the scaling law and sidebranching dynamics shouldexist However previous scaling law did not take sidebranch-ing dynamics into account Therefore it should be morereasonable to characterize CDT by the onset of sidebranchinginstability Furthermore the empirical scaling law indicatesthat beyond CDT increasing the thermal gradient 119866 willenhance the sidebranching dynamics This destabilizingeffect of thermal gradient 119866 on the sidebranching dynamicshas been observed in [7ndash10] Noise amplification theoryfailed in describing the effect of thermal gradient 119866 onsidebranching dynamics [7] Therefore to acquire a deepunderstanding of CDT and sidebranching it is essential toproposemore reasonable physical explanations on the scalinglaw and the destabilizing mechanism
Although sidebranching dynamics has received consid-erable attentions in free dendritic growth [1 4ndash6] only somebasic understanding about the sidebranching dynamics hasbeen obtained in directional solidification [7 10ndash15] Gero-gelin et al [7] presented amodel to describe the sidebranchingdynamics in which the noise amplitude at the cellular tipwas controlled by the feedback of sidebranches However it isdifficult to determine the growth factor in theirmodel On theother hand experimental results and phase field simulationsindicated that the diffusion instability of dendritic trunkis the most possible reason for sidebranching dynamics[14ndash16] and revealed that the initial sidebranching spacingonly depends on the pulling velocity but the sidebranchamplitude is determined not only by pulling velocity 119881 butalso by the primary spacing and the thermal gradient 119866With consideration of the overall effects of primary spacingpulling velocity 119881 and thermal gradient 119866 on the onset ofsidebranches the scaling law of CDTmay gain more physicalfoundation
In this article firstly we briefly review the factors indetermining the onset of sidebranches from previous exper-iments and quantitative phase field simulations Then a newscaling law of CDT is proposed based on the sidebranchingdynamics and the physical foundation of this new scalinglaw is presented The effects of pulling velocity and thermalgradient on CDT will be analyzed according to this newscaling law Finally the quantitative phase field simulationswill be used to testify this proposed scaling law
2 Factors Determining Sidebranches duringDirectional Solidification
It has been widely accepted that the primary spacing 1205821pulling velocity 119881 and thermal gradient 119866 play an importantrole in determining the sidebranches dynamics The effectsof 1205821 119881 and 119866 on the dendritic growth have also been wellstudied Here the related results and how the sidebranchingdynamics is affected by the controlled parameters are recalledbriefly
Primary spacing 1205821 affects the sidebranch dynamicssignificantly In experiments the microstructures near theCDT show dendritic array with large spacing and cellulararray with small spacing [7ndash10] Quantitative phase field
simulations revealed the intrinsic reason [15 17] that thesmaller spacing suppresses the sidebranch growth due to thestrong interdendritic solutal interaction However after theappearance of sidebranch the location of first sidebranchand the initial sidebranch spacing are almost independent ofprimary spacing Therefore the primary spacing only influ-ences the amplitude of the sidebranch but does not change theinitial sidebranch spacing Accordingly the primary spacingonly supplies spacing for sidebranch growth
Pulling velocity 119881 is an important control parameterduring directional solidification According to experimentalinvestigations and simulation the initial sidebranch spacing1205822 has a scaling law with the pulling velocity 1205822infin119881120572 [7 1415 18] Dendrite trunk also depends on the pulling velocitygreatly The width of dendrite trunk decreases as pullingvelocity increases Within the same primary spacing thedecrease of pulling velocity enlarges the diffusion lengthwhich enhances the interdendritic interaction and suppressesthe sidebranch growth As aforementioned the interdendriticinteraction does not change the initial sidebranch spacingTherefore the variation of initial sidebranch spacing ismainlyattributed to the variation of pulling velocity In previousinvestigations the variation of initial sidebranch spacing 1205822with pulling velocity satisfies 1205822infin119881minus059 [7 14 18]
The role of thermal gradient 119866 in sidebranch growthis a little bit complex Previous investigation on the cell-dendrite transition indicated that positive thermal gradientpromotes the generation of sidebranch [7] but the initialsidebranch spacing 1205822 was independent of thermal gradient119866 [14] It is attributed to the remarkable interdendritic solutalinteraction near CDT where the thermal gradient 119866 signif-icantly affects the mush zone in directional solidificationHowever the details on the thermal gradient effects arestill absent and some confusion still exists For examplethe effects of thermal gradient on dendritic morphologiesare relatively weaker for dendritic array growth where thesidebranch amplitude is almost independent of thermalgradient [15]
3 Scaling Law of CDT from Sidebranching
From the above review we can found that previous inves-tigations have presented lots of details about sidebranch-ing dynamics during directional solidification However theintrinsic sidebranching mechanism during directional solid-ification is still not fully revealed Comparedwith free growththe interdendritic interaction will suppress the sidebranchgrowth so the interdendrite interaction plays an importantrole in determining the sidebranches It is still difficult toreveal CDT by directly deriving the sidebranching amplitudeevolution from the basic diffusion equation and interfacecondition
Here we focus on the interdendritic solutal interactionto reveal the CDT and derive the scaling law of CDT inthe following Sketch of the dendrite with sidebranches anddefinition of parameters during the derivations are shownin Figure 1 where 120588119905119894119901 is the dendritic tip radius 119911119905119894119901 is thetip position along 119911-axis 1205821 is primary spacing 1205822 is the
Advances in Condensed Matter Physics 3
Λ 1
Λ 2
2
ztip minus z
1
Figure 1 Sketch of sidebranches and parameters definition in thederivation of scaling law where 1205821 is primary spacing 1205822 is thesidebranch spacing Λ 1 and Λ 2 are groove width and 119911119905119894119901 is the tipposition along 119911-axis
sidebranch spacing and Λ 1 and Λ 2 are groove width Theparameter definitions are as follows solutal diffusion length119897119863 = 119863119881 thermal length 119897119879 = 1198790119866 and capillary length1198890 = Γ1198790 where119863 is the liquid solutal diffusion coefficient1198790 = 1198981198620(119896-1)119896 Γ is the Gibbs-Thomson coefficient [3]1198620 is the initial alloys composition 119898 is the liquid slopeand 119896 is the equilibrium solute-partition coefficient In theinterdendrite region we define the groove width 2Λ andlateral diffusion length 119897119871119863 = 119863119881119871 where 119881119871 is the lateralinterface velocity
Here the CDT is characterized by the appearance ofsidebranches Sidebranches firstly appear near the dendritictip and then are amplified in the groove by the lateral growthof dendritic trunk Two assumptions are adopted in thederivation (1) the occurrence of CDT corresponds with thevisibility of sidebranches (2) the intrinsic conditions forsidebranching are based on the fact that the lateral diffusionaccelerates the lateral instability while the groove widthblocks the lateral instability According to assumption (1)the ratio of the sidebranch amplitude 119860 and the sidebranchwavelength 1205822 can be adopted to characterize the CDTAccording to assumption (2) CDT is determined by thecompetition between the lateral diffusion length 119897119871119863 and thegroove width 2Λ Both 119897119871119863 and 2Λ have the length dimensionso the groove instability can be assumed to be proportionalto the groove width 2Λ and inversely proportional to thelateral diffusion length 119897119871119863 Therefore the two dimension-less characteristics 1198601205822 and Λ119897119871119863 can be the representa-tive of dimensionless characterization and driving force of
sidebranching dynamics respectively For smaller sidebranchamplitude 1198601205822 is a function of Λ119897119871119863
1198601205822 = 119891( Λ
119897119871119863) (1)
Near CDT the lateral diffusion length is very large andthe groove width is small which results in that the Λ119897119871119863approaches zero With Taylor expansion of 1198601205822 asymp 119891(0) +1198911015840(0)Λ119897119871119863+119900((Λ119897119871119863)2) we can assume that the relationshipbetween the driving force and the characterization of thesidebranch is linear near CDT where both 1198601205822 and Λ119897119871119863are very small Then the function in (1) can be approximatedby a linear function 1198601205822 sim Λ119897119871119863 or 119860 sim 1205822Λ119897119871119863 Ifwe define certain finite amplitude 119860 as the representation ofsidebranches appearance then this finite amplitude 119860 leadsto a constant 1205822Λ119897119871119863 at CDT for different pulling velocitiesand thermal gradients
According to the asymptotic analysis of dendrite growthin directional solidification by Spencer and Huppert [19] inthe region of 120588119905119894119901 ≪ 119911119905119894119901 minus 119911 ≪ 119897119879 as shown in Figure 1 wehave
Λ = 1205821 (1 minus 119886 minus radic(119911119905119894119901 minus 119911) 119897119879)2
(2)
where 119886 is a modified parameter to represent the dendritictrunk width in the groove Then within a period of side-branches the lateral interface velocity 119881119871 and the lateraldiffusion length 119897119871119863 can be represented as
119881119871 = (Λ 1 minus Λ 2) (1205822119881 ) (3)
119897119871119863 prop 4119897119863radic119897119879 (119911119905119894119901 minus 119911)1205821
(4)
In experiments the visible amplitude is most likely toappear at 119911 = 119911119905119894119901-1198991205822 where 119899 is a constant [7 8] Accordingto (1)ndash(4) a new simplified scaling law of CDT can bedescribed as
1205821119862119863 sim 120582minus142 1198971198631211989714119879 (1 minus 119886 minus (1198991205822)12 119897minus12119879 )minus12 (5)
In formula (5) sidebranch spacing 1205822 and solutal dif-fusion length 119897119863 = 119863119881 are related to the pulling velocity119881while thermal length 119897119879 = 1198790119866 is related to the thermalgradient 119866 The first three terms on the right side of formula(5) are directly connected to the exponent parameters in thescaling law of 1205821119862119863infin119881120572119866120573 while the contribution of the lastterm (1 minus 120572 minus (1198991205822)12119897minus12119879 )minus12 to the exponent parametersdepends on 1205822119897119879 According to this new scaling law theeffects of pulling velocity 119881 and thermal gradient 119866 on theCDT can be well addressed
To validate this new scaling law the data in [12] is adoptedas a paradigm for further analysis To describe the effect ofthermal length 119897119879 more conveniently we define
119891 (119897119879) = 11989714119879 (1 minus 119886 minus (1198991205822)12 119897minus12119879 )minus12 (6)
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20 40 60 80 100
10
15
20
25
3035
2minus1
4Ｆ ＄
12(Ｆ ４
)
Ｆ４=500m =-049Ｆ４=800m =-044Ｆ４=1400m =-041
V (ms)
Figure 2 The scaling law between the critical primary spacingand the pulling velocity with different thermal lengths in the cell-dendrite transition described by formula (5) where 1205822 asymp 350Vminus06119886=01 and 119899=2
According to the data in [12] the relationship betweensidebranch spacing 1205822 and pulling velocity 119881 is 1205822 asymp350Vminus06 the thermal length 119897119879 is about 500sim1400 120583m andthe parameters of 119886 and 119899 in (6) are 01 and 2 respectively
In the above new scaling law described by formula (5) theexponent parameter 120572 for pulling velocity 119881 is not constantbut is related to the thermal gradient 119866 Figure 2 shows theeffect of 119897119879 on the exponent parameter 120572 When 119897119879 increasesfrom 500 120583m to 1400 120583m 120572 increases from -049 to -041correspondingly It shows that within the range of 119897119879 = 500sim1400 120583m the value of 120572 consists with the scaling law proposedby Gerogelin et al (120572 = minus12) [7] where 120572 = minus046 inuniform cellulardendritic arrayWhen1205822119897119879 is small enoughwe have 1205821119862119863 sim 120582minus142 11989712119863 11989714119879 sim 119881minus035 then 120572 will be -035which agrees with the result of Trivedi et alrsquos experiments[8 9]
As to the exponent parameter 120573 for thermal gradient119866 in the scaling law Figure 3 shows the variation of 119891(119897119879)with 119897119879 When 119897119879 is small the effect of the term (1 minus 119886 minus(1198991205822)12119897minus12119879 )minus12 cannot be overlooked As shown in theinset the power function fitting gives the exponent 120573 asndash0122 which is consistentwith the fitting results inGerogelinet alrsquos experiments (120573 = minus18) [7] As the thermal gradient119866 decreases 119897119879 increases and the variation of term (1 minus 119886 minus(1198991205822)12119897minus12119879 )minus12 makes the exponent parameter 120573 increaseAs shown in Figure 3 large 119897119879 ensures 120573 = minus14 (1205821119862119863 sim120582minus142 11989712119863 11989714119879 sim 119866minus14) Therefore for a small thermal gradientcorresponding to large 119897119879 the exponent parameter of thethermal gradient in the scaling law is very close to Trivedi etalrsquos results [8 9]
The form of this scaling law can return back to theempirical one and reconciles the difference of the exponentparameters in previous experiments Analysis on the expo-nent parameters in the scaling law of 1205821119862119863infin119881120572119866120573 indicates
1000 10000
15
20
25
30
500 600 700 800 900 1000140142144146148150152
f (l T
)
fitting by y = a x^bR^2 = 09949a 652979 plusmn 005445b 012161 plusmn 000126
Ｆ４ (m)
lowast
Figure 3 The effect of the thermal length on the scaling law offormula (5) The inset presents the local region with power lawfitting The corresponding parameters are 1205822 asymp 350Vminus06 with 119881=50120583ms a=01 and n=2
the variation of exponents 120572 and 120573 with different solidifica-tion systems When 1205822119897119879 997888rarr 0 1205821119862119863 sim 120582minus142 11989712119863 11989714119879 sim119881minus035119866minus14 that is 120572=-035 and 120573=-025 which is close toTrivedi et alrsquos experimental results [8 9] Previous experimentalso mentioned that the change of thermal gradient 119866influences the exponent parameters Teng et al [9] pointedout that the increase of thermal gradient 119866 induces a largersystematic deviation from 120572 = 120573 = minus13 Similarly inGerogelin et alrsquos experiments [7] the thermal gradient 119866 wasrelatively large and 120572 = minus05 and 120573 = minus18 This agrees withthe analysis presented here which shows that the exponentparameters vary with thermal gradient 119866 For a large thermalgradient 119866 the exponent parameter may be selected as 120572 =minus045 and 120573 = minus0122 Accordingly the power law presentedhere could settle the argument in previous experiments aboutthe discrepancy of exponent parameters
As to the destabilizing effect of thermal gradient 119866 on thesidebranching dynamics it also can be explained accordingto this new scaling lawThe sidebranching instability is deter-mined by the competition between lateral diffusion length 119897119871119863and the groove width 2Λ Both of 119897119871119863 andΛ decrease with theincrease of thermal gradient119866 Small diffusion length impliesthe enhanced diffusion instability while small groove width ishelpful to stabilize the cell However from (2) (3) and (4) wecan get Λ119897119871119863 sim (1 minus 119886)radic119897119879(119911119905119894119901 minus 119911) minus 1 so Λ119897119871119863 decreaseswith the thermal length 119897119879 that is Λ119897119871119863 increases with thethermal gradient 119866 This indicates that the effect of thermalgradient 119866 on the lateral diffusion length 119897119871119863 predominatesThe destabilizing effect of thermal gradient 119866 comes fromthe decrease of lateral diffusion length 119897119871119863 which is similarto the destabilizing effect of increasing pulling velocity 119881 inplanar instability [1 20 21] So the lateral diffusion length119897119871119863 bridges the thermal gradient 119866 and the sidebranchinginstability which can be used to explain the destabilizingnature of thermal gradient
The scaling law here is related to three parameters thesidebranch spacing the diffusion length and the thermal
Advances in Condensed Matter Physics 5
length which are related to the undercooling and thermalgradient respectively In experiments all these parametersvary in different systems with different parameters Accord-ingly the validity of the scaling law can be well checked withdesigned experiments In this research the new scaling lawwill be validated by a benchmark fromquantitative phase fieldsimulation instead
Note that the noise effects on the sidebranching dynamicsin dendrite growth have been a controversial issue formany years [22] However there is an agreement that thepresence of stochastic noise will not affect the frequency ofsidebranching while the value of the amplitude is a functionof noise intensity level Just as predicted by [23] the transitionof cellular to dendrite growth may be modified by the noiseintensity level In the derivation of the scaling law thesidebranching amplitude is considered as a function of threelength scales that are independent of the noise Thereforethe noise may affect the transition points but the scaling lawbetween these lengths still exists At an adequate noise levelsimilar to that in the experiment the scaling law is valid
4 Validation of the Scaling Law byPhase Field Simulation
The development of phase field method makes it possible toquantitatively investigate microstructure evolution in solidi-fication [18] and the quantitative phase field simulation hasbeen widely used to investigate the sidebranching dynamicsin crystal growth [6 15 16] and the primary spacing selec-tion mechanism in directional solidification [17 24ndash26] Tofurther validate the scaling law the quantitative phase fieldmethod [18] is employed For the directional solidificationsimulation by (7) and (8) assume one-sided diffusion andfrozen temperature approximation in which119879 = 119879119862+119866119911 and119879119862 is the temperature at the cooling end and 119866 is the thermalgradient along 119911-axis The dynamic evolution equations ofphase field model in the moving frame with pulling velocity119881 are
1205910 (1 minus (1 minus 119896) 119911 + (119898119888infin119896) 119866119897119879 ) (120597119905120601 minus 119881120597119911120601)= nabla (119882(120579)2 nabla120601) minus 120597119909 [119882 (120579)1198821015840 (120579) 120597119910120601]+ 120597119910 [119882 (120579)1198821015840 (120579) 120597119909120601] + 120601 minus 1206013 + 120582 (1 minus 1206012)2 (119880
+ 119911 + (119898119888infin119896) 119866119897119879 )
(7)
120597119905119888 minus 119881120597119911119888 = nabla sdot [119863119871 1 minus 1206011 + 119896 minus (1 minus 119896) 120601nabla119888
+ (119863119871 1 minus 1206011 + 119896 minus (1 minus 119896) 120601 + 1198820radic2
(120597119905120601 minus 119881120597119911120601)1003816100381610038161003816nabla1206011003816100381610038161003816 )
sdot 119888 (1 minus 119896)1 + 119896 minus (1 minus 119896) 120601nabla120601]
(8)
with
119882(120579) = 1198820 (1 + 1205744 cos 4120579) (9)
119880 = ((2119896119888119888infin) (1 + 119896 minus (1 minus 119896) 120601) minus 1)(1 minus 119896) (10)
119897119879 = |119898| 119888infin (1 minus 119896)(119896119866) (11)
where 1198820 are the parameters of the interface thickness 1205910is the relaxation time for phase field model and 120582 is thecoupling constant which are related to physical quantitiesby 1198890 = 11988611198820120582 and 1205910 = 119886212058211988220 119863 1205744 is the anisotropicintensity of the surface tension 120579 is the angle between thenormal vector of the interface and the preferred orientation119898 is the liquidus slope k is the partition coefficient and119888infin is the concentration in the far away field Here 1198890 =Γ(119898119888infin(1-119896)119896) 1198861=08839 and 1198862=06267 and Γ is theGibbs-Thomson coefficient
The transparent alloy SCN-043wtC152 is adoptedwhich has been widely used to investigate the evolution ofdendritic pattern [27] The chemical diffusion coefficient ofthe liquid phase is 119863 asymp 045times10minus9 m2s partition coefficient119896=005 Gibbs-Thomson coefficient Γ = 648times10minus8 Km andthe slope of liquidus line 119898=-542 Kmol The surface tensionanisotropy intensity is assumed as 1205744=0005
This quantitative phase field simulation on dendriticgrowth with the presented parameters has been widely per-formed in our previous investigations [15 17]The phase fieldsimulation on the sidebranching indicates that 1205822=150Vminus059and the parameters 119886 and 119899 in (6) are about 04 and 5respectively Then the exponents in the new proposed scalinglaw for the simulation system are 120572 = minus035 and 120573 = minus021according to formula (5) which are different from 120572 = 120573 =minus13 [8 9] and 120572 = minus12 and 120573 = minus18 [7] These threedifferent groups of exponents will be quantitatively examinedaccording to the phase field simulation results
Here a benchmark is designed to directly compare theexponents in different scaling lawsThe exponents are usuallyobtained by fitting plenty of simulation or experiment resultswith different control parameters However heavy workloadand artificial judgment on the onset of sidebranching arerequired by this method In a different way we design abenchmark instead of finding the exponents in the simulationsystem as follows The steady state of interface morphologyin specific primary spacing with onset of sidebranchingis firstly presented where the thermal gradient 1198660 andpulling velocity 1198810 as well as the morphology will be thereferences Then the critical pulling velocity for the CDTcan be extrapolated from different scaling laws along withthe variation of thermal gradient 119866 in the fixed primaryspacing The solidliquid morphologies corresponding tothe different critical velocities are obtained by phase fieldsimulation Finally by comparing the simulation results withthe reference morphology the scaling law with differentexponents is evaluated Here 1198660=202 Kcm and 1198810=20120583ms are chosen as the referential control parameters Thecritical primary spacing for CDT is 1205821=160 120583m and thereferential morphology is shown in Figure 4(a) With 1205821=160
6 Advances in Condensed Matter Physics
(a)
10
10
100
(III)
(II)
(I)
(G 0 V0)
G (Kcm)V
(m
s)
=-13 =-13=-035=-021=-12 =-18
(b)
Figure 4 The critical microstructure of cell-dendritic transition at the benchmark (a) and the criteria of cell-dendrite transition accordingto the different power laws based on the benchmark (b)
V0=28ms V0=37ms V0=45ms V0=77msV0=28ms
Figure 5 The cellulardendritic morphologies with different pulling velocities when 119866=52 Kcm
120583m and 1198660=52 Kcm the different critical pulling velocitiesfor CDT are 119881I=77 120583ms 119881II=45 120583ms and 119881III=28 120583msrespectively for three groups of exponent parameters (-13-13) (-035 -021) and (-12 -18) as shown by the dot inFigure 4(b)
Figure 5 presents the cellulardendrite morphologies fordifferent pulling velocities with 1205821=160 120583m and1198660=52 KcmCompared with the referential morphology in Figure 4(a)the critical pulling velocity is around 35 120583ms The pullingvelocity 77 120583ms predicted by (-13 -13) is obviously largerthan the critical pulling velocity for CDT while the pullingvelocity 28 120583ms predicted by (-12 -18) is close to the criticalpulling velocity but with cellular morphology The pulling
velocity 45 120583ms from the new proposed scaling is also closeto the critical pulling velocity of CDT
The critical pulling velocity of CDT can be further foundwithin higher accuracy In the simplified form of the newproposed scaling law (formula (5)) the contribution of theterm 1205822119897119879 on the critical pulling velocity in (6) is overlookedHere by submitting 1205822=150Vminus059 into (6) the effect of theterm 1205822119897119879 can be revealed Considering the overall effectsof thermal gradient 119866 and pulling velocity 119881 the variationof the left-hand side of formula (5) with pulling velocity fortwo different thermal gradients is presented in Figure 6 Itshows that the exponent 120572 is near -035 with small thermalgradient 119866 However 120572 deviates from -035 gradually as the
Advances in Condensed Matter Physics 7
10 20 30 40 50 609000
12000
15000
18000
21000
24000
27000
G=52Kcm G=202Kcm
2minus1
4Ｆ ＄
12(Ｆ ４
)
V (ms)
Fitting by power law =-038Fitting by power law =-042
Figure 6 The variation of the value of right-hand side in (5) withpulling velocity for different thermal gradients
thermal gradient 119866 increases whichmeans that the influenceof pulling velocity 119881 on 1205822119897119879 has significant impact on119891(119897119879)with relatively small 119897119879 Considering the contribution ofpulling velocity 119881 on 119891(119897119879) Figure 6 gives the critical pullingvelocity of CDT as 37120583ms when 1205821=160 120583m and 1198660=52Kcm in the simulation system The interface morphologyfor 37 120583ms is very close to the reference morphology in thebenchmark
5 Results and Discussion
To summarize a simplified scaling law of CDT duringdirectional solidification is derived with considering thesidebranching dynamics The exponent parameters corre-sponding to the pulling velocity and thermal gradient inthe new scaling law are discussed The analysis shows thatthe exponent parameters in the scaling law vary with dif-ferent solidification systems and reconcile the discrepancyin previous experimental results The form of this scalinglaw can return back to the empirical one and reconciles thedifference of the exponent parameters in previous exper-iments The destabilizing mechanism of thermal gradientin the sidebranching dynamics can be revealed by lateraldiffusion length The new scaling law is also validated bya benchmark from quantitative phase field simulation Theappropriate experimental verification of the scaling law canbe similar to that done by Teng et al [9] With the apparatusone can use different systems and parameters to check thecellular-to-dendrite transition
Furthermore the proposed scaling law is more than theconciliation of the controversy in previous experiments Onone hand compared with previous scaling law the newscaling law is with more physical foundation related to thesidebranching dynamics On the other hand it indicates
that the thermal gradient and pulling velocity are coupledtogether in describing the CDT within a large range param-eter space Only in local parameter space the scaling law ofCDT has a simple form consisted with previous experimentalresults where the thermal gradient and pulling velocity aredecoupled Therefore the proposed scaling law is with moreprecision in predicting CDT in a large range parameter spacecompared with previous investigations
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The work was supported by the fund of the State KeyLaboratory of Solidification Processing in NWPU (Grant noSKLSP201725)
References
[1] J S Langer ldquoInstabilities and pattern formation in crystalgrowthrdquoReviews ofModern Physics vol 52 no 1 pp 1ndash28 1980
[2] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993
[3] W J Boettinger S R Coriell A L Greer et al ldquoSolidificationmicrostructures recent developments future directionsrdquo ActaMaterialia vol 48 no 1 pp 43ndash70 2000
[4] A Karma and W-J Rappel ldquoPhase-field model of dendriticsidebranchingwith thermal noiserdquoPhysical Review E StatisticalPhysics Plasmas Fluids and Related Interdisciplinary Topicsvol 60 no 4 pp 3614ndash3625 1999
[5] M EGlicksman J S Lowengrub SW Li et al ldquoAdeterministicmechanism for dendritic solidification kineticsrdquoThe Journal oftheMinerals Metals ampMaterials Society vol 59 no 8 pp 27ndash342007
[6] Z Wang J Wang and G Yang ldquoPhase-field investigationof effects of surface-tension anisotropy on deterministic side-branching in solutal dendritic growthrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 78 no 4Article ID 042601 2008
[7] M Georgelin and A Pocheau ldquoOnset of sidebranching indirectional solidificationrdquo Physical Review E Statistical PhysicsPlasmas Fluids and Related Interdisciplinary Topics vol 57 no3 pp 3189ndash3203 1998
[8] R Trivedi Y Shen and S Liu ldquoCellular-to-dendritic transitionduring the directional solidification of binary alloysrdquo Metal-lurgical and Materials Transactions A Physical Metallurgy andMaterials Science vol 34 no 2 pp 395ndash401 2003
[9] J Teng S Liu and R Trivedi ldquoOnset of sidewise instabilityand cell-dendrite transition in directional solidificationrdquo ActaMaterialia vol 57 no 12 pp 3497ndash3508 2009
[10] G L Ding On primary dendritic spacing during unidirectionalsolidification [PhD thesis] Northwestern Polytechnical Univer-sity Xirsquoan China 1997
[11] E Acer E Cadırlı H Erol H Kaya and M Gunduz ldquoEffectsof growth rates and compositions on dendrite arm spacings indirectionally solidified Al-Zn alloysrdquoMetallurgical and Materi-als Transactions A Physical Metallurgy and Materials Sciencevol 48 no 12 pp 5911ndash5923 2017
8 Advances in Condensed Matter Physics
[12] A Pocheau S Bodea and M Georgelin ldquoSelf-organized den-dritic sidebranching in directional solidification sidebranchcoherence within uncorrelated burstsrdquo Physical Review E Sta-tistical Nonlinear and SoftMatter Physics vol 80 no 3 ArticleID 031601 2009
[13] J S Kirkaldy L X Liu and A Kroupa ldquoThin film forcedvelocity cells and cellular dendrites-II Analysis of datardquo ActaMetallurgica et Materialia vol 43 no 8 pp 2905ndash2915 1995
[14] K Somboonsuk J T Mason and R Trivedi ldquoInterdendriticspacing part I Experimental studiesrdquo Metallurgical Transac-tions A Physical Metallurgy and Materials Science vol 15 no6 pp 967ndash975 1984
[15] Z JWang J CWang andGC Yang ldquoPhase field investigationon the selection of initial sidebranch spacing in directionalsolidificationrdquo IOP Conference Series Materials Science andEngineering vol 27 no 1 p 012009 2012
[16] B Echebarria A Karma and S Gurevich ldquoOnset of side-branching in directional solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 81 no 2Article ID 021608 2010
[17] Z Wang J Li J Wang and Y Zhou ldquoPhase field modelingthe selection mechanism of primary dendritic spacing indirectional solidificationrdquo Acta Materialia vol 60 no 5 pp1957ndash1964 2012
[18] B Echebarria R Folch A Karma and M Plapp ldquoQuantitativephase-field model of alloy solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 70 no 6Article ID 061604 2004
[19] B J Spencer andH EHuppert ldquoRelationship between dendritetip characteristics and dendrite spacings in alloy directionalsolidificationrdquo Journal of Crystal Growth vol 200 no 1-2 pp287ndash296 1999
[20] Z Wang J Wang and G Yang ldquoOnset of initial planarinstability with surface-tension anisotropy during directionalsolidificationrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 80 no 5 Article ID 052603 2009
[21] Z Wang J Wang and G Yang ldquoFourier synthesis predictingonset of the initial instability during directional solidificationrdquoApplied Physics Letters vol 94 no 6 p 061920 2009
[22] J J Xu Interfacial Wave Theory of Pattern Formation in Solidi-fication Dendrites Fingers Cells and Free Boundary SpringerSeries in Synergetics Springer International Publishing 2ndedition 2017
[23] G Agez M G Clerc E Louvergneaux and R G RojasldquoBifurcations of emerging patterns in the presence of additivenoiserdquo Physical Review E Statistical Nonlinear and Soft MatterPhysics vol 87 no 4 Article ID 042919 2013
[24] I Steinbach ldquoEffect of interface anisotropy on spacing selectionin constrained dendrite growthrdquoActaMaterialia vol 56 no 18pp 4965ndash4971 2008
[25] S Gurevich M Amoorezaei and N Provatas ldquoPhase-fieldstudy of spacing evolution during transient growthrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 82no 5 Article ID 051606 2010
[26] M Amoorezaei S Gurevich and N Provatas ldquoSpacing char-acterization in Al-Cu alloys directionally solidified under tran-sient growth conditionsrdquo Acta Materialia vol 58 no 18 pp6115ndash6124 2010
[27] W Losert B Q Shi and H Z Cummins ldquoEvolution ofdendritic patterns during alloy solidification onset of the initialinstabilityrdquo Proceedings of the National Acadamy of Sciences ofthe United States of America vol 95 no 2 pp 431ndash438 1998
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2 Advances in Condensed Matter Physics
of the typical characteristics in CDT and the connectionsbetween the scaling law and sidebranching dynamics shouldexist However previous scaling law did not take sidebranch-ing dynamics into account Therefore it should be morereasonable to characterize CDT by the onset of sidebranchinginstability Furthermore the empirical scaling law indicatesthat beyond CDT increasing the thermal gradient 119866 willenhance the sidebranching dynamics This destabilizingeffect of thermal gradient 119866 on the sidebranching dynamicshas been observed in [7ndash10] Noise amplification theoryfailed in describing the effect of thermal gradient 119866 onsidebranching dynamics [7] Therefore to acquire a deepunderstanding of CDT and sidebranching it is essential toproposemore reasonable physical explanations on the scalinglaw and the destabilizing mechanism
Although sidebranching dynamics has received consid-erable attentions in free dendritic growth [1 4ndash6] only somebasic understanding about the sidebranching dynamics hasbeen obtained in directional solidification [7 10ndash15] Gero-gelin et al [7] presented amodel to describe the sidebranchingdynamics in which the noise amplitude at the cellular tipwas controlled by the feedback of sidebranches However it isdifficult to determine the growth factor in theirmodel On theother hand experimental results and phase field simulationsindicated that the diffusion instability of dendritic trunkis the most possible reason for sidebranching dynamics[14ndash16] and revealed that the initial sidebranching spacingonly depends on the pulling velocity but the sidebranchamplitude is determined not only by pulling velocity 119881 butalso by the primary spacing and the thermal gradient 119866With consideration of the overall effects of primary spacingpulling velocity 119881 and thermal gradient 119866 on the onset ofsidebranches the scaling law of CDTmay gain more physicalfoundation
In this article firstly we briefly review the factors indetermining the onset of sidebranches from previous exper-iments and quantitative phase field simulations Then a newscaling law of CDT is proposed based on the sidebranchingdynamics and the physical foundation of this new scalinglaw is presented The effects of pulling velocity and thermalgradient on CDT will be analyzed according to this newscaling law Finally the quantitative phase field simulationswill be used to testify this proposed scaling law
2 Factors Determining Sidebranches duringDirectional Solidification
It has been widely accepted that the primary spacing 1205821pulling velocity 119881 and thermal gradient 119866 play an importantrole in determining the sidebranches dynamics The effectsof 1205821 119881 and 119866 on the dendritic growth have also been wellstudied Here the related results and how the sidebranchingdynamics is affected by the controlled parameters are recalledbriefly
Primary spacing 1205821 affects the sidebranch dynamicssignificantly In experiments the microstructures near theCDT show dendritic array with large spacing and cellulararray with small spacing [7ndash10] Quantitative phase field
simulations revealed the intrinsic reason [15 17] that thesmaller spacing suppresses the sidebranch growth due to thestrong interdendritic solutal interaction However after theappearance of sidebranch the location of first sidebranchand the initial sidebranch spacing are almost independent ofprimary spacing Therefore the primary spacing only influ-ences the amplitude of the sidebranch but does not change theinitial sidebranch spacing Accordingly the primary spacingonly supplies spacing for sidebranch growth
Pulling velocity 119881 is an important control parameterduring directional solidification According to experimentalinvestigations and simulation the initial sidebranch spacing1205822 has a scaling law with the pulling velocity 1205822infin119881120572 [7 1415 18] Dendrite trunk also depends on the pulling velocitygreatly The width of dendrite trunk decreases as pullingvelocity increases Within the same primary spacing thedecrease of pulling velocity enlarges the diffusion lengthwhich enhances the interdendritic interaction and suppressesthe sidebranch growth As aforementioned the interdendriticinteraction does not change the initial sidebranch spacingTherefore the variation of initial sidebranch spacing ismainlyattributed to the variation of pulling velocity In previousinvestigations the variation of initial sidebranch spacing 1205822with pulling velocity satisfies 1205822infin119881minus059 [7 14 18]
The role of thermal gradient 119866 in sidebranch growthis a little bit complex Previous investigation on the cell-dendrite transition indicated that positive thermal gradientpromotes the generation of sidebranch [7] but the initialsidebranch spacing 1205822 was independent of thermal gradient119866 [14] It is attributed to the remarkable interdendritic solutalinteraction near CDT where the thermal gradient 119866 signif-icantly affects the mush zone in directional solidificationHowever the details on the thermal gradient effects arestill absent and some confusion still exists For examplethe effects of thermal gradient on dendritic morphologiesare relatively weaker for dendritic array growth where thesidebranch amplitude is almost independent of thermalgradient [15]
3 Scaling Law of CDT from Sidebranching
From the above review we can found that previous inves-tigations have presented lots of details about sidebranch-ing dynamics during directional solidification However theintrinsic sidebranching mechanism during directional solid-ification is still not fully revealed Comparedwith free growththe interdendritic interaction will suppress the sidebranchgrowth so the interdendrite interaction plays an importantrole in determining the sidebranches It is still difficult toreveal CDT by directly deriving the sidebranching amplitudeevolution from the basic diffusion equation and interfacecondition
Here we focus on the interdendritic solutal interactionto reveal the CDT and derive the scaling law of CDT inthe following Sketch of the dendrite with sidebranches anddefinition of parameters during the derivations are shownin Figure 1 where 120588119905119894119901 is the dendritic tip radius 119911119905119894119901 is thetip position along 119911-axis 1205821 is primary spacing 1205822 is the
Advances in Condensed Matter Physics 3
Λ 1
Λ 2
2
ztip minus z
1
Figure 1 Sketch of sidebranches and parameters definition in thederivation of scaling law where 1205821 is primary spacing 1205822 is thesidebranch spacing Λ 1 and Λ 2 are groove width and 119911119905119894119901 is the tipposition along 119911-axis
sidebranch spacing and Λ 1 and Λ 2 are groove width Theparameter definitions are as follows solutal diffusion length119897119863 = 119863119881 thermal length 119897119879 = 1198790119866 and capillary length1198890 = Γ1198790 where119863 is the liquid solutal diffusion coefficient1198790 = 1198981198620(119896-1)119896 Γ is the Gibbs-Thomson coefficient [3]1198620 is the initial alloys composition 119898 is the liquid slopeand 119896 is the equilibrium solute-partition coefficient In theinterdendrite region we define the groove width 2Λ andlateral diffusion length 119897119871119863 = 119863119881119871 where 119881119871 is the lateralinterface velocity
Here the CDT is characterized by the appearance ofsidebranches Sidebranches firstly appear near the dendritictip and then are amplified in the groove by the lateral growthof dendritic trunk Two assumptions are adopted in thederivation (1) the occurrence of CDT corresponds with thevisibility of sidebranches (2) the intrinsic conditions forsidebranching are based on the fact that the lateral diffusionaccelerates the lateral instability while the groove widthblocks the lateral instability According to assumption (1)the ratio of the sidebranch amplitude 119860 and the sidebranchwavelength 1205822 can be adopted to characterize the CDTAccording to assumption (2) CDT is determined by thecompetition between the lateral diffusion length 119897119871119863 and thegroove width 2Λ Both 119897119871119863 and 2Λ have the length dimensionso the groove instability can be assumed to be proportionalto the groove width 2Λ and inversely proportional to thelateral diffusion length 119897119871119863 Therefore the two dimension-less characteristics 1198601205822 and Λ119897119871119863 can be the representa-tive of dimensionless characterization and driving force of
sidebranching dynamics respectively For smaller sidebranchamplitude 1198601205822 is a function of Λ119897119871119863
1198601205822 = 119891( Λ
119897119871119863) (1)
Near CDT the lateral diffusion length is very large andthe groove width is small which results in that the Λ119897119871119863approaches zero With Taylor expansion of 1198601205822 asymp 119891(0) +1198911015840(0)Λ119897119871119863+119900((Λ119897119871119863)2) we can assume that the relationshipbetween the driving force and the characterization of thesidebranch is linear near CDT where both 1198601205822 and Λ119897119871119863are very small Then the function in (1) can be approximatedby a linear function 1198601205822 sim Λ119897119871119863 or 119860 sim 1205822Λ119897119871119863 Ifwe define certain finite amplitude 119860 as the representation ofsidebranches appearance then this finite amplitude 119860 leadsto a constant 1205822Λ119897119871119863 at CDT for different pulling velocitiesand thermal gradients
According to the asymptotic analysis of dendrite growthin directional solidification by Spencer and Huppert [19] inthe region of 120588119905119894119901 ≪ 119911119905119894119901 minus 119911 ≪ 119897119879 as shown in Figure 1 wehave
Λ = 1205821 (1 minus 119886 minus radic(119911119905119894119901 minus 119911) 119897119879)2
(2)
where 119886 is a modified parameter to represent the dendritictrunk width in the groove Then within a period of side-branches the lateral interface velocity 119881119871 and the lateraldiffusion length 119897119871119863 can be represented as
119881119871 = (Λ 1 minus Λ 2) (1205822119881 ) (3)
119897119871119863 prop 4119897119863radic119897119879 (119911119905119894119901 minus 119911)1205821
(4)
In experiments the visible amplitude is most likely toappear at 119911 = 119911119905119894119901-1198991205822 where 119899 is a constant [7 8] Accordingto (1)ndash(4) a new simplified scaling law of CDT can bedescribed as
1205821119862119863 sim 120582minus142 1198971198631211989714119879 (1 minus 119886 minus (1198991205822)12 119897minus12119879 )minus12 (5)
In formula (5) sidebranch spacing 1205822 and solutal dif-fusion length 119897119863 = 119863119881 are related to the pulling velocity119881while thermal length 119897119879 = 1198790119866 is related to the thermalgradient 119866 The first three terms on the right side of formula(5) are directly connected to the exponent parameters in thescaling law of 1205821119862119863infin119881120572119866120573 while the contribution of the lastterm (1 minus 120572 minus (1198991205822)12119897minus12119879 )minus12 to the exponent parametersdepends on 1205822119897119879 According to this new scaling law theeffects of pulling velocity 119881 and thermal gradient 119866 on theCDT can be well addressed
To validate this new scaling law the data in [12] is adoptedas a paradigm for further analysis To describe the effect ofthermal length 119897119879 more conveniently we define
119891 (119897119879) = 11989714119879 (1 minus 119886 minus (1198991205822)12 119897minus12119879 )minus12 (6)
4 Advances in Condensed Matter Physics
20 40 60 80 100
10
15
20
25
3035
2minus1
4Ｆ ＄
12(Ｆ ４
)
Ｆ４=500m =-049Ｆ４=800m =-044Ｆ４=1400m =-041
V (ms)
Figure 2 The scaling law between the critical primary spacingand the pulling velocity with different thermal lengths in the cell-dendrite transition described by formula (5) where 1205822 asymp 350Vminus06119886=01 and 119899=2
According to the data in [12] the relationship betweensidebranch spacing 1205822 and pulling velocity 119881 is 1205822 asymp350Vminus06 the thermal length 119897119879 is about 500sim1400 120583m andthe parameters of 119886 and 119899 in (6) are 01 and 2 respectively
In the above new scaling law described by formula (5) theexponent parameter 120572 for pulling velocity 119881 is not constantbut is related to the thermal gradient 119866 Figure 2 shows theeffect of 119897119879 on the exponent parameter 120572 When 119897119879 increasesfrom 500 120583m to 1400 120583m 120572 increases from -049 to -041correspondingly It shows that within the range of 119897119879 = 500sim1400 120583m the value of 120572 consists with the scaling law proposedby Gerogelin et al (120572 = minus12) [7] where 120572 = minus046 inuniform cellulardendritic arrayWhen1205822119897119879 is small enoughwe have 1205821119862119863 sim 120582minus142 11989712119863 11989714119879 sim 119881minus035 then 120572 will be -035which agrees with the result of Trivedi et alrsquos experiments[8 9]
As to the exponent parameter 120573 for thermal gradient119866 in the scaling law Figure 3 shows the variation of 119891(119897119879)with 119897119879 When 119897119879 is small the effect of the term (1 minus 119886 minus(1198991205822)12119897minus12119879 )minus12 cannot be overlooked As shown in theinset the power function fitting gives the exponent 120573 asndash0122 which is consistentwith the fitting results inGerogelinet alrsquos experiments (120573 = minus18) [7] As the thermal gradient119866 decreases 119897119879 increases and the variation of term (1 minus 119886 minus(1198991205822)12119897minus12119879 )minus12 makes the exponent parameter 120573 increaseAs shown in Figure 3 large 119897119879 ensures 120573 = minus14 (1205821119862119863 sim120582minus142 11989712119863 11989714119879 sim 119866minus14) Therefore for a small thermal gradientcorresponding to large 119897119879 the exponent parameter of thethermal gradient in the scaling law is very close to Trivedi etalrsquos results [8 9]
The form of this scaling law can return back to theempirical one and reconciles the difference of the exponentparameters in previous experiments Analysis on the expo-nent parameters in the scaling law of 1205821119862119863infin119881120572119866120573 indicates
1000 10000
15
20
25
30
500 600 700 800 900 1000140142144146148150152
f (l T
)
fitting by y = a x^bR^2 = 09949a 652979 plusmn 005445b 012161 plusmn 000126
Ｆ４ (m)
lowast
Figure 3 The effect of the thermal length on the scaling law offormula (5) The inset presents the local region with power lawfitting The corresponding parameters are 1205822 asymp 350Vminus06 with 119881=50120583ms a=01 and n=2
the variation of exponents 120572 and 120573 with different solidifica-tion systems When 1205822119897119879 997888rarr 0 1205821119862119863 sim 120582minus142 11989712119863 11989714119879 sim119881minus035119866minus14 that is 120572=-035 and 120573=-025 which is close toTrivedi et alrsquos experimental results [8 9] Previous experimentalso mentioned that the change of thermal gradient 119866influences the exponent parameters Teng et al [9] pointedout that the increase of thermal gradient 119866 induces a largersystematic deviation from 120572 = 120573 = minus13 Similarly inGerogelin et alrsquos experiments [7] the thermal gradient 119866 wasrelatively large and 120572 = minus05 and 120573 = minus18 This agrees withthe analysis presented here which shows that the exponentparameters vary with thermal gradient 119866 For a large thermalgradient 119866 the exponent parameter may be selected as 120572 =minus045 and 120573 = minus0122 Accordingly the power law presentedhere could settle the argument in previous experiments aboutthe discrepancy of exponent parameters
As to the destabilizing effect of thermal gradient 119866 on thesidebranching dynamics it also can be explained accordingto this new scaling lawThe sidebranching instability is deter-mined by the competition between lateral diffusion length 119897119871119863and the groove width 2Λ Both of 119897119871119863 andΛ decrease with theincrease of thermal gradient119866 Small diffusion length impliesthe enhanced diffusion instability while small groove width ishelpful to stabilize the cell However from (2) (3) and (4) wecan get Λ119897119871119863 sim (1 minus 119886)radic119897119879(119911119905119894119901 minus 119911) minus 1 so Λ119897119871119863 decreaseswith the thermal length 119897119879 that is Λ119897119871119863 increases with thethermal gradient 119866 This indicates that the effect of thermalgradient 119866 on the lateral diffusion length 119897119871119863 predominatesThe destabilizing effect of thermal gradient 119866 comes fromthe decrease of lateral diffusion length 119897119871119863 which is similarto the destabilizing effect of increasing pulling velocity 119881 inplanar instability [1 20 21] So the lateral diffusion length119897119871119863 bridges the thermal gradient 119866 and the sidebranchinginstability which can be used to explain the destabilizingnature of thermal gradient
The scaling law here is related to three parameters thesidebranch spacing the diffusion length and the thermal
Advances in Condensed Matter Physics 5
length which are related to the undercooling and thermalgradient respectively In experiments all these parametersvary in different systems with different parameters Accord-ingly the validity of the scaling law can be well checked withdesigned experiments In this research the new scaling lawwill be validated by a benchmark fromquantitative phase fieldsimulation instead
Note that the noise effects on the sidebranching dynamicsin dendrite growth have been a controversial issue formany years [22] However there is an agreement that thepresence of stochastic noise will not affect the frequency ofsidebranching while the value of the amplitude is a functionof noise intensity level Just as predicted by [23] the transitionof cellular to dendrite growth may be modified by the noiseintensity level In the derivation of the scaling law thesidebranching amplitude is considered as a function of threelength scales that are independent of the noise Thereforethe noise may affect the transition points but the scaling lawbetween these lengths still exists At an adequate noise levelsimilar to that in the experiment the scaling law is valid
4 Validation of the Scaling Law byPhase Field Simulation
The development of phase field method makes it possible toquantitatively investigate microstructure evolution in solidi-fication [18] and the quantitative phase field simulation hasbeen widely used to investigate the sidebranching dynamicsin crystal growth [6 15 16] and the primary spacing selec-tion mechanism in directional solidification [17 24ndash26] Tofurther validate the scaling law the quantitative phase fieldmethod [18] is employed For the directional solidificationsimulation by (7) and (8) assume one-sided diffusion andfrozen temperature approximation in which119879 = 119879119862+119866119911 and119879119862 is the temperature at the cooling end and 119866 is the thermalgradient along 119911-axis The dynamic evolution equations ofphase field model in the moving frame with pulling velocity119881 are
1205910 (1 minus (1 minus 119896) 119911 + (119898119888infin119896) 119866119897119879 ) (120597119905120601 minus 119881120597119911120601)= nabla (119882(120579)2 nabla120601) minus 120597119909 [119882 (120579)1198821015840 (120579) 120597119910120601]+ 120597119910 [119882 (120579)1198821015840 (120579) 120597119909120601] + 120601 minus 1206013 + 120582 (1 minus 1206012)2 (119880
+ 119911 + (119898119888infin119896) 119866119897119879 )
(7)
120597119905119888 minus 119881120597119911119888 = nabla sdot [119863119871 1 minus 1206011 + 119896 minus (1 minus 119896) 120601nabla119888
+ (119863119871 1 minus 1206011 + 119896 minus (1 minus 119896) 120601 + 1198820radic2
(120597119905120601 minus 119881120597119911120601)1003816100381610038161003816nabla1206011003816100381610038161003816 )
sdot 119888 (1 minus 119896)1 + 119896 minus (1 minus 119896) 120601nabla120601]
(8)
with
119882(120579) = 1198820 (1 + 1205744 cos 4120579) (9)
119880 = ((2119896119888119888infin) (1 + 119896 minus (1 minus 119896) 120601) minus 1)(1 minus 119896) (10)
119897119879 = |119898| 119888infin (1 minus 119896)(119896119866) (11)
where 1198820 are the parameters of the interface thickness 1205910is the relaxation time for phase field model and 120582 is thecoupling constant which are related to physical quantitiesby 1198890 = 11988611198820120582 and 1205910 = 119886212058211988220 119863 1205744 is the anisotropicintensity of the surface tension 120579 is the angle between thenormal vector of the interface and the preferred orientation119898 is the liquidus slope k is the partition coefficient and119888infin is the concentration in the far away field Here 1198890 =Γ(119898119888infin(1-119896)119896) 1198861=08839 and 1198862=06267 and Γ is theGibbs-Thomson coefficient
The transparent alloy SCN-043wtC152 is adoptedwhich has been widely used to investigate the evolution ofdendritic pattern [27] The chemical diffusion coefficient ofthe liquid phase is 119863 asymp 045times10minus9 m2s partition coefficient119896=005 Gibbs-Thomson coefficient Γ = 648times10minus8 Km andthe slope of liquidus line 119898=-542 Kmol The surface tensionanisotropy intensity is assumed as 1205744=0005
This quantitative phase field simulation on dendriticgrowth with the presented parameters has been widely per-formed in our previous investigations [15 17]The phase fieldsimulation on the sidebranching indicates that 1205822=150Vminus059and the parameters 119886 and 119899 in (6) are about 04 and 5respectively Then the exponents in the new proposed scalinglaw for the simulation system are 120572 = minus035 and 120573 = minus021according to formula (5) which are different from 120572 = 120573 =minus13 [8 9] and 120572 = minus12 and 120573 = minus18 [7] These threedifferent groups of exponents will be quantitatively examinedaccording to the phase field simulation results
Here a benchmark is designed to directly compare theexponents in different scaling lawsThe exponents are usuallyobtained by fitting plenty of simulation or experiment resultswith different control parameters However heavy workloadand artificial judgment on the onset of sidebranching arerequired by this method In a different way we design abenchmark instead of finding the exponents in the simulationsystem as follows The steady state of interface morphologyin specific primary spacing with onset of sidebranchingis firstly presented where the thermal gradient 1198660 andpulling velocity 1198810 as well as the morphology will be thereferences Then the critical pulling velocity for the CDTcan be extrapolated from different scaling laws along withthe variation of thermal gradient 119866 in the fixed primaryspacing The solidliquid morphologies corresponding tothe different critical velocities are obtained by phase fieldsimulation Finally by comparing the simulation results withthe reference morphology the scaling law with differentexponents is evaluated Here 1198660=202 Kcm and 1198810=20120583ms are chosen as the referential control parameters Thecritical primary spacing for CDT is 1205821=160 120583m and thereferential morphology is shown in Figure 4(a) With 1205821=160
6 Advances in Condensed Matter Physics
(a)
10
10
100
(III)
(II)
(I)
(G 0 V0)
G (Kcm)V
(m
s)
=-13 =-13=-035=-021=-12 =-18
(b)
Figure 4 The critical microstructure of cell-dendritic transition at the benchmark (a) and the criteria of cell-dendrite transition accordingto the different power laws based on the benchmark (b)
V0=28ms V0=37ms V0=45ms V0=77msV0=28ms
Figure 5 The cellulardendritic morphologies with different pulling velocities when 119866=52 Kcm
120583m and 1198660=52 Kcm the different critical pulling velocitiesfor CDT are 119881I=77 120583ms 119881II=45 120583ms and 119881III=28 120583msrespectively for three groups of exponent parameters (-13-13) (-035 -021) and (-12 -18) as shown by the dot inFigure 4(b)
Figure 5 presents the cellulardendrite morphologies fordifferent pulling velocities with 1205821=160 120583m and1198660=52 KcmCompared with the referential morphology in Figure 4(a)the critical pulling velocity is around 35 120583ms The pullingvelocity 77 120583ms predicted by (-13 -13) is obviously largerthan the critical pulling velocity for CDT while the pullingvelocity 28 120583ms predicted by (-12 -18) is close to the criticalpulling velocity but with cellular morphology The pulling
velocity 45 120583ms from the new proposed scaling is also closeto the critical pulling velocity of CDT
The critical pulling velocity of CDT can be further foundwithin higher accuracy In the simplified form of the newproposed scaling law (formula (5)) the contribution of theterm 1205822119897119879 on the critical pulling velocity in (6) is overlookedHere by submitting 1205822=150Vminus059 into (6) the effect of theterm 1205822119897119879 can be revealed Considering the overall effectsof thermal gradient 119866 and pulling velocity 119881 the variationof the left-hand side of formula (5) with pulling velocity fortwo different thermal gradients is presented in Figure 6 Itshows that the exponent 120572 is near -035 with small thermalgradient 119866 However 120572 deviates from -035 gradually as the
Advances in Condensed Matter Physics 7
10 20 30 40 50 609000
12000
15000
18000
21000
24000
27000
G=52Kcm G=202Kcm
2minus1
4Ｆ ＄
12(Ｆ ４
)
V (ms)
Fitting by power law =-038Fitting by power law =-042
Figure 6 The variation of the value of right-hand side in (5) withpulling velocity for different thermal gradients
thermal gradient 119866 increases whichmeans that the influenceof pulling velocity 119881 on 1205822119897119879 has significant impact on119891(119897119879)with relatively small 119897119879 Considering the contribution ofpulling velocity 119881 on 119891(119897119879) Figure 6 gives the critical pullingvelocity of CDT as 37120583ms when 1205821=160 120583m and 1198660=52Kcm in the simulation system The interface morphologyfor 37 120583ms is very close to the reference morphology in thebenchmark
5 Results and Discussion
To summarize a simplified scaling law of CDT duringdirectional solidification is derived with considering thesidebranching dynamics The exponent parameters corre-sponding to the pulling velocity and thermal gradient inthe new scaling law are discussed The analysis shows thatthe exponent parameters in the scaling law vary with dif-ferent solidification systems and reconcile the discrepancyin previous experimental results The form of this scalinglaw can return back to the empirical one and reconciles thedifference of the exponent parameters in previous exper-iments The destabilizing mechanism of thermal gradientin the sidebranching dynamics can be revealed by lateraldiffusion length The new scaling law is also validated bya benchmark from quantitative phase field simulation Theappropriate experimental verification of the scaling law canbe similar to that done by Teng et al [9] With the apparatusone can use different systems and parameters to check thecellular-to-dendrite transition
Furthermore the proposed scaling law is more than theconciliation of the controversy in previous experiments Onone hand compared with previous scaling law the newscaling law is with more physical foundation related to thesidebranching dynamics On the other hand it indicates
that the thermal gradient and pulling velocity are coupledtogether in describing the CDT within a large range param-eter space Only in local parameter space the scaling law ofCDT has a simple form consisted with previous experimentalresults where the thermal gradient and pulling velocity aredecoupled Therefore the proposed scaling law is with moreprecision in predicting CDT in a large range parameter spacecompared with previous investigations
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The work was supported by the fund of the State KeyLaboratory of Solidification Processing in NWPU (Grant noSKLSP201725)
References
[1] J S Langer ldquoInstabilities and pattern formation in crystalgrowthrdquoReviews ofModern Physics vol 52 no 1 pp 1ndash28 1980
[2] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993
[3] W J Boettinger S R Coriell A L Greer et al ldquoSolidificationmicrostructures recent developments future directionsrdquo ActaMaterialia vol 48 no 1 pp 43ndash70 2000
[4] A Karma and W-J Rappel ldquoPhase-field model of dendriticsidebranchingwith thermal noiserdquoPhysical Review E StatisticalPhysics Plasmas Fluids and Related Interdisciplinary Topicsvol 60 no 4 pp 3614ndash3625 1999
[5] M EGlicksman J S Lowengrub SW Li et al ldquoAdeterministicmechanism for dendritic solidification kineticsrdquoThe Journal oftheMinerals Metals ampMaterials Society vol 59 no 8 pp 27ndash342007
[6] Z Wang J Wang and G Yang ldquoPhase-field investigationof effects of surface-tension anisotropy on deterministic side-branching in solutal dendritic growthrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 78 no 4Article ID 042601 2008
[7] M Georgelin and A Pocheau ldquoOnset of sidebranching indirectional solidificationrdquo Physical Review E Statistical PhysicsPlasmas Fluids and Related Interdisciplinary Topics vol 57 no3 pp 3189ndash3203 1998
[8] R Trivedi Y Shen and S Liu ldquoCellular-to-dendritic transitionduring the directional solidification of binary alloysrdquo Metal-lurgical and Materials Transactions A Physical Metallurgy andMaterials Science vol 34 no 2 pp 395ndash401 2003
[9] J Teng S Liu and R Trivedi ldquoOnset of sidewise instabilityand cell-dendrite transition in directional solidificationrdquo ActaMaterialia vol 57 no 12 pp 3497ndash3508 2009
[10] G L Ding On primary dendritic spacing during unidirectionalsolidification [PhD thesis] Northwestern Polytechnical Univer-sity Xirsquoan China 1997
[11] E Acer E Cadırlı H Erol H Kaya and M Gunduz ldquoEffectsof growth rates and compositions on dendrite arm spacings indirectionally solidified Al-Zn alloysrdquoMetallurgical and Materi-als Transactions A Physical Metallurgy and Materials Sciencevol 48 no 12 pp 5911ndash5923 2017
8 Advances in Condensed Matter Physics
[12] A Pocheau S Bodea and M Georgelin ldquoSelf-organized den-dritic sidebranching in directional solidification sidebranchcoherence within uncorrelated burstsrdquo Physical Review E Sta-tistical Nonlinear and SoftMatter Physics vol 80 no 3 ArticleID 031601 2009
[13] J S Kirkaldy L X Liu and A Kroupa ldquoThin film forcedvelocity cells and cellular dendrites-II Analysis of datardquo ActaMetallurgica et Materialia vol 43 no 8 pp 2905ndash2915 1995
[14] K Somboonsuk J T Mason and R Trivedi ldquoInterdendriticspacing part I Experimental studiesrdquo Metallurgical Transac-tions A Physical Metallurgy and Materials Science vol 15 no6 pp 967ndash975 1984
[15] Z JWang J CWang andGC Yang ldquoPhase field investigationon the selection of initial sidebranch spacing in directionalsolidificationrdquo IOP Conference Series Materials Science andEngineering vol 27 no 1 p 012009 2012
[16] B Echebarria A Karma and S Gurevich ldquoOnset of side-branching in directional solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 81 no 2Article ID 021608 2010
[17] Z Wang J Li J Wang and Y Zhou ldquoPhase field modelingthe selection mechanism of primary dendritic spacing indirectional solidificationrdquo Acta Materialia vol 60 no 5 pp1957ndash1964 2012
[18] B Echebarria R Folch A Karma and M Plapp ldquoQuantitativephase-field model of alloy solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 70 no 6Article ID 061604 2004
[19] B J Spencer andH EHuppert ldquoRelationship between dendritetip characteristics and dendrite spacings in alloy directionalsolidificationrdquo Journal of Crystal Growth vol 200 no 1-2 pp287ndash296 1999
[20] Z Wang J Wang and G Yang ldquoOnset of initial planarinstability with surface-tension anisotropy during directionalsolidificationrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 80 no 5 Article ID 052603 2009
[21] Z Wang J Wang and G Yang ldquoFourier synthesis predictingonset of the initial instability during directional solidificationrdquoApplied Physics Letters vol 94 no 6 p 061920 2009
[22] J J Xu Interfacial Wave Theory of Pattern Formation in Solidi-fication Dendrites Fingers Cells and Free Boundary SpringerSeries in Synergetics Springer International Publishing 2ndedition 2017
[23] G Agez M G Clerc E Louvergneaux and R G RojasldquoBifurcations of emerging patterns in the presence of additivenoiserdquo Physical Review E Statistical Nonlinear and Soft MatterPhysics vol 87 no 4 Article ID 042919 2013
[24] I Steinbach ldquoEffect of interface anisotropy on spacing selectionin constrained dendrite growthrdquoActaMaterialia vol 56 no 18pp 4965ndash4971 2008
[25] S Gurevich M Amoorezaei and N Provatas ldquoPhase-fieldstudy of spacing evolution during transient growthrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 82no 5 Article ID 051606 2010
[26] M Amoorezaei S Gurevich and N Provatas ldquoSpacing char-acterization in Al-Cu alloys directionally solidified under tran-sient growth conditionsrdquo Acta Materialia vol 58 no 18 pp6115ndash6124 2010
[27] W Losert B Q Shi and H Z Cummins ldquoEvolution ofdendritic patterns during alloy solidification onset of the initialinstabilityrdquo Proceedings of the National Acadamy of Sciences ofthe United States of America vol 95 no 2 pp 431ndash438 1998
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Advances in Condensed Matter Physics 3
Λ 1
Λ 2
2
ztip minus z
1
Figure 1 Sketch of sidebranches and parameters definition in thederivation of scaling law where 1205821 is primary spacing 1205822 is thesidebranch spacing Λ 1 and Λ 2 are groove width and 119911119905119894119901 is the tipposition along 119911-axis
sidebranch spacing and Λ 1 and Λ 2 are groove width Theparameter definitions are as follows solutal diffusion length119897119863 = 119863119881 thermal length 119897119879 = 1198790119866 and capillary length1198890 = Γ1198790 where119863 is the liquid solutal diffusion coefficient1198790 = 1198981198620(119896-1)119896 Γ is the Gibbs-Thomson coefficient [3]1198620 is the initial alloys composition 119898 is the liquid slopeand 119896 is the equilibrium solute-partition coefficient In theinterdendrite region we define the groove width 2Λ andlateral diffusion length 119897119871119863 = 119863119881119871 where 119881119871 is the lateralinterface velocity
Here the CDT is characterized by the appearance ofsidebranches Sidebranches firstly appear near the dendritictip and then are amplified in the groove by the lateral growthof dendritic trunk Two assumptions are adopted in thederivation (1) the occurrence of CDT corresponds with thevisibility of sidebranches (2) the intrinsic conditions forsidebranching are based on the fact that the lateral diffusionaccelerates the lateral instability while the groove widthblocks the lateral instability According to assumption (1)the ratio of the sidebranch amplitude 119860 and the sidebranchwavelength 1205822 can be adopted to characterize the CDTAccording to assumption (2) CDT is determined by thecompetition between the lateral diffusion length 119897119871119863 and thegroove width 2Λ Both 119897119871119863 and 2Λ have the length dimensionso the groove instability can be assumed to be proportionalto the groove width 2Λ and inversely proportional to thelateral diffusion length 119897119871119863 Therefore the two dimension-less characteristics 1198601205822 and Λ119897119871119863 can be the representa-tive of dimensionless characterization and driving force of
sidebranching dynamics respectively For smaller sidebranchamplitude 1198601205822 is a function of Λ119897119871119863
1198601205822 = 119891( Λ
119897119871119863) (1)
Near CDT the lateral diffusion length is very large andthe groove width is small which results in that the Λ119897119871119863approaches zero With Taylor expansion of 1198601205822 asymp 119891(0) +1198911015840(0)Λ119897119871119863+119900((Λ119897119871119863)2) we can assume that the relationshipbetween the driving force and the characterization of thesidebranch is linear near CDT where both 1198601205822 and Λ119897119871119863are very small Then the function in (1) can be approximatedby a linear function 1198601205822 sim Λ119897119871119863 or 119860 sim 1205822Λ119897119871119863 Ifwe define certain finite amplitude 119860 as the representation ofsidebranches appearance then this finite amplitude 119860 leadsto a constant 1205822Λ119897119871119863 at CDT for different pulling velocitiesand thermal gradients
According to the asymptotic analysis of dendrite growthin directional solidification by Spencer and Huppert [19] inthe region of 120588119905119894119901 ≪ 119911119905119894119901 minus 119911 ≪ 119897119879 as shown in Figure 1 wehave
Λ = 1205821 (1 minus 119886 minus radic(119911119905119894119901 minus 119911) 119897119879)2
(2)
where 119886 is a modified parameter to represent the dendritictrunk width in the groove Then within a period of side-branches the lateral interface velocity 119881119871 and the lateraldiffusion length 119897119871119863 can be represented as
119881119871 = (Λ 1 minus Λ 2) (1205822119881 ) (3)
119897119871119863 prop 4119897119863radic119897119879 (119911119905119894119901 minus 119911)1205821
(4)
In experiments the visible amplitude is most likely toappear at 119911 = 119911119905119894119901-1198991205822 where 119899 is a constant [7 8] Accordingto (1)ndash(4) a new simplified scaling law of CDT can bedescribed as
1205821119862119863 sim 120582minus142 1198971198631211989714119879 (1 minus 119886 minus (1198991205822)12 119897minus12119879 )minus12 (5)
In formula (5) sidebranch spacing 1205822 and solutal dif-fusion length 119897119863 = 119863119881 are related to the pulling velocity119881while thermal length 119897119879 = 1198790119866 is related to the thermalgradient 119866 The first three terms on the right side of formula(5) are directly connected to the exponent parameters in thescaling law of 1205821119862119863infin119881120572119866120573 while the contribution of the lastterm (1 minus 120572 minus (1198991205822)12119897minus12119879 )minus12 to the exponent parametersdepends on 1205822119897119879 According to this new scaling law theeffects of pulling velocity 119881 and thermal gradient 119866 on theCDT can be well addressed
To validate this new scaling law the data in [12] is adoptedas a paradigm for further analysis To describe the effect ofthermal length 119897119879 more conveniently we define
119891 (119897119879) = 11989714119879 (1 minus 119886 minus (1198991205822)12 119897minus12119879 )minus12 (6)
4 Advances in Condensed Matter Physics
20 40 60 80 100
10
15
20
25
3035
2minus1
4Ｆ ＄
12(Ｆ ４
)
Ｆ４=500m =-049Ｆ４=800m =-044Ｆ４=1400m =-041
V (ms)
Figure 2 The scaling law between the critical primary spacingand the pulling velocity with different thermal lengths in the cell-dendrite transition described by formula (5) where 1205822 asymp 350Vminus06119886=01 and 119899=2
According to the data in [12] the relationship betweensidebranch spacing 1205822 and pulling velocity 119881 is 1205822 asymp350Vminus06 the thermal length 119897119879 is about 500sim1400 120583m andthe parameters of 119886 and 119899 in (6) are 01 and 2 respectively
In the above new scaling law described by formula (5) theexponent parameter 120572 for pulling velocity 119881 is not constantbut is related to the thermal gradient 119866 Figure 2 shows theeffect of 119897119879 on the exponent parameter 120572 When 119897119879 increasesfrom 500 120583m to 1400 120583m 120572 increases from -049 to -041correspondingly It shows that within the range of 119897119879 = 500sim1400 120583m the value of 120572 consists with the scaling law proposedby Gerogelin et al (120572 = minus12) [7] where 120572 = minus046 inuniform cellulardendritic arrayWhen1205822119897119879 is small enoughwe have 1205821119862119863 sim 120582minus142 11989712119863 11989714119879 sim 119881minus035 then 120572 will be -035which agrees with the result of Trivedi et alrsquos experiments[8 9]
As to the exponent parameter 120573 for thermal gradient119866 in the scaling law Figure 3 shows the variation of 119891(119897119879)with 119897119879 When 119897119879 is small the effect of the term (1 minus 119886 minus(1198991205822)12119897minus12119879 )minus12 cannot be overlooked As shown in theinset the power function fitting gives the exponent 120573 asndash0122 which is consistentwith the fitting results inGerogelinet alrsquos experiments (120573 = minus18) [7] As the thermal gradient119866 decreases 119897119879 increases and the variation of term (1 minus 119886 minus(1198991205822)12119897minus12119879 )minus12 makes the exponent parameter 120573 increaseAs shown in Figure 3 large 119897119879 ensures 120573 = minus14 (1205821119862119863 sim120582minus142 11989712119863 11989714119879 sim 119866minus14) Therefore for a small thermal gradientcorresponding to large 119897119879 the exponent parameter of thethermal gradient in the scaling law is very close to Trivedi etalrsquos results [8 9]
The form of this scaling law can return back to theempirical one and reconciles the difference of the exponentparameters in previous experiments Analysis on the expo-nent parameters in the scaling law of 1205821119862119863infin119881120572119866120573 indicates
1000 10000
15
20
25
30
500 600 700 800 900 1000140142144146148150152
f (l T
)
fitting by y = a x^bR^2 = 09949a 652979 plusmn 005445b 012161 plusmn 000126
Ｆ４ (m)
lowast
Figure 3 The effect of the thermal length on the scaling law offormula (5) The inset presents the local region with power lawfitting The corresponding parameters are 1205822 asymp 350Vminus06 with 119881=50120583ms a=01 and n=2
the variation of exponents 120572 and 120573 with different solidifica-tion systems When 1205822119897119879 997888rarr 0 1205821119862119863 sim 120582minus142 11989712119863 11989714119879 sim119881minus035119866minus14 that is 120572=-035 and 120573=-025 which is close toTrivedi et alrsquos experimental results [8 9] Previous experimentalso mentioned that the change of thermal gradient 119866influences the exponent parameters Teng et al [9] pointedout that the increase of thermal gradient 119866 induces a largersystematic deviation from 120572 = 120573 = minus13 Similarly inGerogelin et alrsquos experiments [7] the thermal gradient 119866 wasrelatively large and 120572 = minus05 and 120573 = minus18 This agrees withthe analysis presented here which shows that the exponentparameters vary with thermal gradient 119866 For a large thermalgradient 119866 the exponent parameter may be selected as 120572 =minus045 and 120573 = minus0122 Accordingly the power law presentedhere could settle the argument in previous experiments aboutthe discrepancy of exponent parameters
As to the destabilizing effect of thermal gradient 119866 on thesidebranching dynamics it also can be explained accordingto this new scaling lawThe sidebranching instability is deter-mined by the competition between lateral diffusion length 119897119871119863and the groove width 2Λ Both of 119897119871119863 andΛ decrease with theincrease of thermal gradient119866 Small diffusion length impliesthe enhanced diffusion instability while small groove width ishelpful to stabilize the cell However from (2) (3) and (4) wecan get Λ119897119871119863 sim (1 minus 119886)radic119897119879(119911119905119894119901 minus 119911) minus 1 so Λ119897119871119863 decreaseswith the thermal length 119897119879 that is Λ119897119871119863 increases with thethermal gradient 119866 This indicates that the effect of thermalgradient 119866 on the lateral diffusion length 119897119871119863 predominatesThe destabilizing effect of thermal gradient 119866 comes fromthe decrease of lateral diffusion length 119897119871119863 which is similarto the destabilizing effect of increasing pulling velocity 119881 inplanar instability [1 20 21] So the lateral diffusion length119897119871119863 bridges the thermal gradient 119866 and the sidebranchinginstability which can be used to explain the destabilizingnature of thermal gradient
The scaling law here is related to three parameters thesidebranch spacing the diffusion length and the thermal
Advances in Condensed Matter Physics 5
length which are related to the undercooling and thermalgradient respectively In experiments all these parametersvary in different systems with different parameters Accord-ingly the validity of the scaling law can be well checked withdesigned experiments In this research the new scaling lawwill be validated by a benchmark fromquantitative phase fieldsimulation instead
Note that the noise effects on the sidebranching dynamicsin dendrite growth have been a controversial issue formany years [22] However there is an agreement that thepresence of stochastic noise will not affect the frequency ofsidebranching while the value of the amplitude is a functionof noise intensity level Just as predicted by [23] the transitionof cellular to dendrite growth may be modified by the noiseintensity level In the derivation of the scaling law thesidebranching amplitude is considered as a function of threelength scales that are independent of the noise Thereforethe noise may affect the transition points but the scaling lawbetween these lengths still exists At an adequate noise levelsimilar to that in the experiment the scaling law is valid
4 Validation of the Scaling Law byPhase Field Simulation
The development of phase field method makes it possible toquantitatively investigate microstructure evolution in solidi-fication [18] and the quantitative phase field simulation hasbeen widely used to investigate the sidebranching dynamicsin crystal growth [6 15 16] and the primary spacing selec-tion mechanism in directional solidification [17 24ndash26] Tofurther validate the scaling law the quantitative phase fieldmethod [18] is employed For the directional solidificationsimulation by (7) and (8) assume one-sided diffusion andfrozen temperature approximation in which119879 = 119879119862+119866119911 and119879119862 is the temperature at the cooling end and 119866 is the thermalgradient along 119911-axis The dynamic evolution equations ofphase field model in the moving frame with pulling velocity119881 are
1205910 (1 minus (1 minus 119896) 119911 + (119898119888infin119896) 119866119897119879 ) (120597119905120601 minus 119881120597119911120601)= nabla (119882(120579)2 nabla120601) minus 120597119909 [119882 (120579)1198821015840 (120579) 120597119910120601]+ 120597119910 [119882 (120579)1198821015840 (120579) 120597119909120601] + 120601 minus 1206013 + 120582 (1 minus 1206012)2 (119880
+ 119911 + (119898119888infin119896) 119866119897119879 )
(7)
120597119905119888 minus 119881120597119911119888 = nabla sdot [119863119871 1 minus 1206011 + 119896 minus (1 minus 119896) 120601nabla119888
+ (119863119871 1 minus 1206011 + 119896 minus (1 minus 119896) 120601 + 1198820radic2
(120597119905120601 minus 119881120597119911120601)1003816100381610038161003816nabla1206011003816100381610038161003816 )
sdot 119888 (1 minus 119896)1 + 119896 minus (1 minus 119896) 120601nabla120601]
(8)
with
119882(120579) = 1198820 (1 + 1205744 cos 4120579) (9)
119880 = ((2119896119888119888infin) (1 + 119896 minus (1 minus 119896) 120601) minus 1)(1 minus 119896) (10)
119897119879 = |119898| 119888infin (1 minus 119896)(119896119866) (11)
where 1198820 are the parameters of the interface thickness 1205910is the relaxation time for phase field model and 120582 is thecoupling constant which are related to physical quantitiesby 1198890 = 11988611198820120582 and 1205910 = 119886212058211988220 119863 1205744 is the anisotropicintensity of the surface tension 120579 is the angle between thenormal vector of the interface and the preferred orientation119898 is the liquidus slope k is the partition coefficient and119888infin is the concentration in the far away field Here 1198890 =Γ(119898119888infin(1-119896)119896) 1198861=08839 and 1198862=06267 and Γ is theGibbs-Thomson coefficient
The transparent alloy SCN-043wtC152 is adoptedwhich has been widely used to investigate the evolution ofdendritic pattern [27] The chemical diffusion coefficient ofthe liquid phase is 119863 asymp 045times10minus9 m2s partition coefficient119896=005 Gibbs-Thomson coefficient Γ = 648times10minus8 Km andthe slope of liquidus line 119898=-542 Kmol The surface tensionanisotropy intensity is assumed as 1205744=0005
This quantitative phase field simulation on dendriticgrowth with the presented parameters has been widely per-formed in our previous investigations [15 17]The phase fieldsimulation on the sidebranching indicates that 1205822=150Vminus059and the parameters 119886 and 119899 in (6) are about 04 and 5respectively Then the exponents in the new proposed scalinglaw for the simulation system are 120572 = minus035 and 120573 = minus021according to formula (5) which are different from 120572 = 120573 =minus13 [8 9] and 120572 = minus12 and 120573 = minus18 [7] These threedifferent groups of exponents will be quantitatively examinedaccording to the phase field simulation results
Here a benchmark is designed to directly compare theexponents in different scaling lawsThe exponents are usuallyobtained by fitting plenty of simulation or experiment resultswith different control parameters However heavy workloadand artificial judgment on the onset of sidebranching arerequired by this method In a different way we design abenchmark instead of finding the exponents in the simulationsystem as follows The steady state of interface morphologyin specific primary spacing with onset of sidebranchingis firstly presented where the thermal gradient 1198660 andpulling velocity 1198810 as well as the morphology will be thereferences Then the critical pulling velocity for the CDTcan be extrapolated from different scaling laws along withthe variation of thermal gradient 119866 in the fixed primaryspacing The solidliquid morphologies corresponding tothe different critical velocities are obtained by phase fieldsimulation Finally by comparing the simulation results withthe reference morphology the scaling law with differentexponents is evaluated Here 1198660=202 Kcm and 1198810=20120583ms are chosen as the referential control parameters Thecritical primary spacing for CDT is 1205821=160 120583m and thereferential morphology is shown in Figure 4(a) With 1205821=160
6 Advances in Condensed Matter Physics
(a)
10
10
100
(III)
(II)
(I)
(G 0 V0)
G (Kcm)V
(m
s)
=-13 =-13=-035=-021=-12 =-18
(b)
Figure 4 The critical microstructure of cell-dendritic transition at the benchmark (a) and the criteria of cell-dendrite transition accordingto the different power laws based on the benchmark (b)
V0=28ms V0=37ms V0=45ms V0=77msV0=28ms
Figure 5 The cellulardendritic morphologies with different pulling velocities when 119866=52 Kcm
120583m and 1198660=52 Kcm the different critical pulling velocitiesfor CDT are 119881I=77 120583ms 119881II=45 120583ms and 119881III=28 120583msrespectively for three groups of exponent parameters (-13-13) (-035 -021) and (-12 -18) as shown by the dot inFigure 4(b)
Figure 5 presents the cellulardendrite morphologies fordifferent pulling velocities with 1205821=160 120583m and1198660=52 KcmCompared with the referential morphology in Figure 4(a)the critical pulling velocity is around 35 120583ms The pullingvelocity 77 120583ms predicted by (-13 -13) is obviously largerthan the critical pulling velocity for CDT while the pullingvelocity 28 120583ms predicted by (-12 -18) is close to the criticalpulling velocity but with cellular morphology The pulling
velocity 45 120583ms from the new proposed scaling is also closeto the critical pulling velocity of CDT
The critical pulling velocity of CDT can be further foundwithin higher accuracy In the simplified form of the newproposed scaling law (formula (5)) the contribution of theterm 1205822119897119879 on the critical pulling velocity in (6) is overlookedHere by submitting 1205822=150Vminus059 into (6) the effect of theterm 1205822119897119879 can be revealed Considering the overall effectsof thermal gradient 119866 and pulling velocity 119881 the variationof the left-hand side of formula (5) with pulling velocity fortwo different thermal gradients is presented in Figure 6 Itshows that the exponent 120572 is near -035 with small thermalgradient 119866 However 120572 deviates from -035 gradually as the
Advances in Condensed Matter Physics 7
10 20 30 40 50 609000
12000
15000
18000
21000
24000
27000
G=52Kcm G=202Kcm
2minus1
4Ｆ ＄
12(Ｆ ４
)
V (ms)
Fitting by power law =-038Fitting by power law =-042
Figure 6 The variation of the value of right-hand side in (5) withpulling velocity for different thermal gradients
thermal gradient 119866 increases whichmeans that the influenceof pulling velocity 119881 on 1205822119897119879 has significant impact on119891(119897119879)with relatively small 119897119879 Considering the contribution ofpulling velocity 119881 on 119891(119897119879) Figure 6 gives the critical pullingvelocity of CDT as 37120583ms when 1205821=160 120583m and 1198660=52Kcm in the simulation system The interface morphologyfor 37 120583ms is very close to the reference morphology in thebenchmark
5 Results and Discussion
To summarize a simplified scaling law of CDT duringdirectional solidification is derived with considering thesidebranching dynamics The exponent parameters corre-sponding to the pulling velocity and thermal gradient inthe new scaling law are discussed The analysis shows thatthe exponent parameters in the scaling law vary with dif-ferent solidification systems and reconcile the discrepancyin previous experimental results The form of this scalinglaw can return back to the empirical one and reconciles thedifference of the exponent parameters in previous exper-iments The destabilizing mechanism of thermal gradientin the sidebranching dynamics can be revealed by lateraldiffusion length The new scaling law is also validated bya benchmark from quantitative phase field simulation Theappropriate experimental verification of the scaling law canbe similar to that done by Teng et al [9] With the apparatusone can use different systems and parameters to check thecellular-to-dendrite transition
Furthermore the proposed scaling law is more than theconciliation of the controversy in previous experiments Onone hand compared with previous scaling law the newscaling law is with more physical foundation related to thesidebranching dynamics On the other hand it indicates
that the thermal gradient and pulling velocity are coupledtogether in describing the CDT within a large range param-eter space Only in local parameter space the scaling law ofCDT has a simple form consisted with previous experimentalresults where the thermal gradient and pulling velocity aredecoupled Therefore the proposed scaling law is with moreprecision in predicting CDT in a large range parameter spacecompared with previous investigations
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The work was supported by the fund of the State KeyLaboratory of Solidification Processing in NWPU (Grant noSKLSP201725)
References
[1] J S Langer ldquoInstabilities and pattern formation in crystalgrowthrdquoReviews ofModern Physics vol 52 no 1 pp 1ndash28 1980
[2] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993
[3] W J Boettinger S R Coriell A L Greer et al ldquoSolidificationmicrostructures recent developments future directionsrdquo ActaMaterialia vol 48 no 1 pp 43ndash70 2000
[4] A Karma and W-J Rappel ldquoPhase-field model of dendriticsidebranchingwith thermal noiserdquoPhysical Review E StatisticalPhysics Plasmas Fluids and Related Interdisciplinary Topicsvol 60 no 4 pp 3614ndash3625 1999
[5] M EGlicksman J S Lowengrub SW Li et al ldquoAdeterministicmechanism for dendritic solidification kineticsrdquoThe Journal oftheMinerals Metals ampMaterials Society vol 59 no 8 pp 27ndash342007
[6] Z Wang J Wang and G Yang ldquoPhase-field investigationof effects of surface-tension anisotropy on deterministic side-branching in solutal dendritic growthrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 78 no 4Article ID 042601 2008
[7] M Georgelin and A Pocheau ldquoOnset of sidebranching indirectional solidificationrdquo Physical Review E Statistical PhysicsPlasmas Fluids and Related Interdisciplinary Topics vol 57 no3 pp 3189ndash3203 1998
[8] R Trivedi Y Shen and S Liu ldquoCellular-to-dendritic transitionduring the directional solidification of binary alloysrdquo Metal-lurgical and Materials Transactions A Physical Metallurgy andMaterials Science vol 34 no 2 pp 395ndash401 2003
[9] J Teng S Liu and R Trivedi ldquoOnset of sidewise instabilityand cell-dendrite transition in directional solidificationrdquo ActaMaterialia vol 57 no 12 pp 3497ndash3508 2009
[10] G L Ding On primary dendritic spacing during unidirectionalsolidification [PhD thesis] Northwestern Polytechnical Univer-sity Xirsquoan China 1997
[11] E Acer E Cadırlı H Erol H Kaya and M Gunduz ldquoEffectsof growth rates and compositions on dendrite arm spacings indirectionally solidified Al-Zn alloysrdquoMetallurgical and Materi-als Transactions A Physical Metallurgy and Materials Sciencevol 48 no 12 pp 5911ndash5923 2017
8 Advances in Condensed Matter Physics
[12] A Pocheau S Bodea and M Georgelin ldquoSelf-organized den-dritic sidebranching in directional solidification sidebranchcoherence within uncorrelated burstsrdquo Physical Review E Sta-tistical Nonlinear and SoftMatter Physics vol 80 no 3 ArticleID 031601 2009
[13] J S Kirkaldy L X Liu and A Kroupa ldquoThin film forcedvelocity cells and cellular dendrites-II Analysis of datardquo ActaMetallurgica et Materialia vol 43 no 8 pp 2905ndash2915 1995
[14] K Somboonsuk J T Mason and R Trivedi ldquoInterdendriticspacing part I Experimental studiesrdquo Metallurgical Transac-tions A Physical Metallurgy and Materials Science vol 15 no6 pp 967ndash975 1984
[15] Z JWang J CWang andGC Yang ldquoPhase field investigationon the selection of initial sidebranch spacing in directionalsolidificationrdquo IOP Conference Series Materials Science andEngineering vol 27 no 1 p 012009 2012
[16] B Echebarria A Karma and S Gurevich ldquoOnset of side-branching in directional solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 81 no 2Article ID 021608 2010
[17] Z Wang J Li J Wang and Y Zhou ldquoPhase field modelingthe selection mechanism of primary dendritic spacing indirectional solidificationrdquo Acta Materialia vol 60 no 5 pp1957ndash1964 2012
[18] B Echebarria R Folch A Karma and M Plapp ldquoQuantitativephase-field model of alloy solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 70 no 6Article ID 061604 2004
[19] B J Spencer andH EHuppert ldquoRelationship between dendritetip characteristics and dendrite spacings in alloy directionalsolidificationrdquo Journal of Crystal Growth vol 200 no 1-2 pp287ndash296 1999
[20] Z Wang J Wang and G Yang ldquoOnset of initial planarinstability with surface-tension anisotropy during directionalsolidificationrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 80 no 5 Article ID 052603 2009
[21] Z Wang J Wang and G Yang ldquoFourier synthesis predictingonset of the initial instability during directional solidificationrdquoApplied Physics Letters vol 94 no 6 p 061920 2009
[22] J J Xu Interfacial Wave Theory of Pattern Formation in Solidi-fication Dendrites Fingers Cells and Free Boundary SpringerSeries in Synergetics Springer International Publishing 2ndedition 2017
[23] G Agez M G Clerc E Louvergneaux and R G RojasldquoBifurcations of emerging patterns in the presence of additivenoiserdquo Physical Review E Statistical Nonlinear and Soft MatterPhysics vol 87 no 4 Article ID 042919 2013
[24] I Steinbach ldquoEffect of interface anisotropy on spacing selectionin constrained dendrite growthrdquoActaMaterialia vol 56 no 18pp 4965ndash4971 2008
[25] S Gurevich M Amoorezaei and N Provatas ldquoPhase-fieldstudy of spacing evolution during transient growthrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 82no 5 Article ID 051606 2010
[26] M Amoorezaei S Gurevich and N Provatas ldquoSpacing char-acterization in Al-Cu alloys directionally solidified under tran-sient growth conditionsrdquo Acta Materialia vol 58 no 18 pp6115ndash6124 2010
[27] W Losert B Q Shi and H Z Cummins ldquoEvolution ofdendritic patterns during alloy solidification onset of the initialinstabilityrdquo Proceedings of the National Acadamy of Sciences ofthe United States of America vol 95 no 2 pp 431ndash438 1998
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Submit your manuscripts atwwwhindawicom
4 Advances in Condensed Matter Physics
20 40 60 80 100
10
15
20
25
3035
2minus1
4Ｆ ＄
12(Ｆ ４
)
Ｆ４=500m =-049Ｆ４=800m =-044Ｆ４=1400m =-041
V (ms)
Figure 2 The scaling law between the critical primary spacingand the pulling velocity with different thermal lengths in the cell-dendrite transition described by formula (5) where 1205822 asymp 350Vminus06119886=01 and 119899=2
According to the data in [12] the relationship betweensidebranch spacing 1205822 and pulling velocity 119881 is 1205822 asymp350Vminus06 the thermal length 119897119879 is about 500sim1400 120583m andthe parameters of 119886 and 119899 in (6) are 01 and 2 respectively
In the above new scaling law described by formula (5) theexponent parameter 120572 for pulling velocity 119881 is not constantbut is related to the thermal gradient 119866 Figure 2 shows theeffect of 119897119879 on the exponent parameter 120572 When 119897119879 increasesfrom 500 120583m to 1400 120583m 120572 increases from -049 to -041correspondingly It shows that within the range of 119897119879 = 500sim1400 120583m the value of 120572 consists with the scaling law proposedby Gerogelin et al (120572 = minus12) [7] where 120572 = minus046 inuniform cellulardendritic arrayWhen1205822119897119879 is small enoughwe have 1205821119862119863 sim 120582minus142 11989712119863 11989714119879 sim 119881minus035 then 120572 will be -035which agrees with the result of Trivedi et alrsquos experiments[8 9]
As to the exponent parameter 120573 for thermal gradient119866 in the scaling law Figure 3 shows the variation of 119891(119897119879)with 119897119879 When 119897119879 is small the effect of the term (1 minus 119886 minus(1198991205822)12119897minus12119879 )minus12 cannot be overlooked As shown in theinset the power function fitting gives the exponent 120573 asndash0122 which is consistentwith the fitting results inGerogelinet alrsquos experiments (120573 = minus18) [7] As the thermal gradient119866 decreases 119897119879 increases and the variation of term (1 minus 119886 minus(1198991205822)12119897minus12119879 )minus12 makes the exponent parameter 120573 increaseAs shown in Figure 3 large 119897119879 ensures 120573 = minus14 (1205821119862119863 sim120582minus142 11989712119863 11989714119879 sim 119866minus14) Therefore for a small thermal gradientcorresponding to large 119897119879 the exponent parameter of thethermal gradient in the scaling law is very close to Trivedi etalrsquos results [8 9]
The form of this scaling law can return back to theempirical one and reconciles the difference of the exponentparameters in previous experiments Analysis on the expo-nent parameters in the scaling law of 1205821119862119863infin119881120572119866120573 indicates
1000 10000
15
20
25
30
500 600 700 800 900 1000140142144146148150152
f (l T
)
fitting by y = a x^bR^2 = 09949a 652979 plusmn 005445b 012161 plusmn 000126
Ｆ４ (m)
lowast
Figure 3 The effect of the thermal length on the scaling law offormula (5) The inset presents the local region with power lawfitting The corresponding parameters are 1205822 asymp 350Vminus06 with 119881=50120583ms a=01 and n=2
the variation of exponents 120572 and 120573 with different solidifica-tion systems When 1205822119897119879 997888rarr 0 1205821119862119863 sim 120582minus142 11989712119863 11989714119879 sim119881minus035119866minus14 that is 120572=-035 and 120573=-025 which is close toTrivedi et alrsquos experimental results [8 9] Previous experimentalso mentioned that the change of thermal gradient 119866influences the exponent parameters Teng et al [9] pointedout that the increase of thermal gradient 119866 induces a largersystematic deviation from 120572 = 120573 = minus13 Similarly inGerogelin et alrsquos experiments [7] the thermal gradient 119866 wasrelatively large and 120572 = minus05 and 120573 = minus18 This agrees withthe analysis presented here which shows that the exponentparameters vary with thermal gradient 119866 For a large thermalgradient 119866 the exponent parameter may be selected as 120572 =minus045 and 120573 = minus0122 Accordingly the power law presentedhere could settle the argument in previous experiments aboutthe discrepancy of exponent parameters
As to the destabilizing effect of thermal gradient 119866 on thesidebranching dynamics it also can be explained accordingto this new scaling lawThe sidebranching instability is deter-mined by the competition between lateral diffusion length 119897119871119863and the groove width 2Λ Both of 119897119871119863 andΛ decrease with theincrease of thermal gradient119866 Small diffusion length impliesthe enhanced diffusion instability while small groove width ishelpful to stabilize the cell However from (2) (3) and (4) wecan get Λ119897119871119863 sim (1 minus 119886)radic119897119879(119911119905119894119901 minus 119911) minus 1 so Λ119897119871119863 decreaseswith the thermal length 119897119879 that is Λ119897119871119863 increases with thethermal gradient 119866 This indicates that the effect of thermalgradient 119866 on the lateral diffusion length 119897119871119863 predominatesThe destabilizing effect of thermal gradient 119866 comes fromthe decrease of lateral diffusion length 119897119871119863 which is similarto the destabilizing effect of increasing pulling velocity 119881 inplanar instability [1 20 21] So the lateral diffusion length119897119871119863 bridges the thermal gradient 119866 and the sidebranchinginstability which can be used to explain the destabilizingnature of thermal gradient
The scaling law here is related to three parameters thesidebranch spacing the diffusion length and the thermal
Advances in Condensed Matter Physics 5
length which are related to the undercooling and thermalgradient respectively In experiments all these parametersvary in different systems with different parameters Accord-ingly the validity of the scaling law can be well checked withdesigned experiments In this research the new scaling lawwill be validated by a benchmark fromquantitative phase fieldsimulation instead
Note that the noise effects on the sidebranching dynamicsin dendrite growth have been a controversial issue formany years [22] However there is an agreement that thepresence of stochastic noise will not affect the frequency ofsidebranching while the value of the amplitude is a functionof noise intensity level Just as predicted by [23] the transitionof cellular to dendrite growth may be modified by the noiseintensity level In the derivation of the scaling law thesidebranching amplitude is considered as a function of threelength scales that are independent of the noise Thereforethe noise may affect the transition points but the scaling lawbetween these lengths still exists At an adequate noise levelsimilar to that in the experiment the scaling law is valid
4 Validation of the Scaling Law byPhase Field Simulation
The development of phase field method makes it possible toquantitatively investigate microstructure evolution in solidi-fication [18] and the quantitative phase field simulation hasbeen widely used to investigate the sidebranching dynamicsin crystal growth [6 15 16] and the primary spacing selec-tion mechanism in directional solidification [17 24ndash26] Tofurther validate the scaling law the quantitative phase fieldmethod [18] is employed For the directional solidificationsimulation by (7) and (8) assume one-sided diffusion andfrozen temperature approximation in which119879 = 119879119862+119866119911 and119879119862 is the temperature at the cooling end and 119866 is the thermalgradient along 119911-axis The dynamic evolution equations ofphase field model in the moving frame with pulling velocity119881 are
1205910 (1 minus (1 minus 119896) 119911 + (119898119888infin119896) 119866119897119879 ) (120597119905120601 minus 119881120597119911120601)= nabla (119882(120579)2 nabla120601) minus 120597119909 [119882 (120579)1198821015840 (120579) 120597119910120601]+ 120597119910 [119882 (120579)1198821015840 (120579) 120597119909120601] + 120601 minus 1206013 + 120582 (1 minus 1206012)2 (119880
+ 119911 + (119898119888infin119896) 119866119897119879 )
(7)
120597119905119888 minus 119881120597119911119888 = nabla sdot [119863119871 1 minus 1206011 + 119896 minus (1 minus 119896) 120601nabla119888
+ (119863119871 1 minus 1206011 + 119896 minus (1 minus 119896) 120601 + 1198820radic2
(120597119905120601 minus 119881120597119911120601)1003816100381610038161003816nabla1206011003816100381610038161003816 )
sdot 119888 (1 minus 119896)1 + 119896 minus (1 minus 119896) 120601nabla120601]
(8)
with
119882(120579) = 1198820 (1 + 1205744 cos 4120579) (9)
119880 = ((2119896119888119888infin) (1 + 119896 minus (1 minus 119896) 120601) minus 1)(1 minus 119896) (10)
119897119879 = |119898| 119888infin (1 minus 119896)(119896119866) (11)
where 1198820 are the parameters of the interface thickness 1205910is the relaxation time for phase field model and 120582 is thecoupling constant which are related to physical quantitiesby 1198890 = 11988611198820120582 and 1205910 = 119886212058211988220 119863 1205744 is the anisotropicintensity of the surface tension 120579 is the angle between thenormal vector of the interface and the preferred orientation119898 is the liquidus slope k is the partition coefficient and119888infin is the concentration in the far away field Here 1198890 =Γ(119898119888infin(1-119896)119896) 1198861=08839 and 1198862=06267 and Γ is theGibbs-Thomson coefficient
The transparent alloy SCN-043wtC152 is adoptedwhich has been widely used to investigate the evolution ofdendritic pattern [27] The chemical diffusion coefficient ofthe liquid phase is 119863 asymp 045times10minus9 m2s partition coefficient119896=005 Gibbs-Thomson coefficient Γ = 648times10minus8 Km andthe slope of liquidus line 119898=-542 Kmol The surface tensionanisotropy intensity is assumed as 1205744=0005
This quantitative phase field simulation on dendriticgrowth with the presented parameters has been widely per-formed in our previous investigations [15 17]The phase fieldsimulation on the sidebranching indicates that 1205822=150Vminus059and the parameters 119886 and 119899 in (6) are about 04 and 5respectively Then the exponents in the new proposed scalinglaw for the simulation system are 120572 = minus035 and 120573 = minus021according to formula (5) which are different from 120572 = 120573 =minus13 [8 9] and 120572 = minus12 and 120573 = minus18 [7] These threedifferent groups of exponents will be quantitatively examinedaccording to the phase field simulation results
Here a benchmark is designed to directly compare theexponents in different scaling lawsThe exponents are usuallyobtained by fitting plenty of simulation or experiment resultswith different control parameters However heavy workloadand artificial judgment on the onset of sidebranching arerequired by this method In a different way we design abenchmark instead of finding the exponents in the simulationsystem as follows The steady state of interface morphologyin specific primary spacing with onset of sidebranchingis firstly presented where the thermal gradient 1198660 andpulling velocity 1198810 as well as the morphology will be thereferences Then the critical pulling velocity for the CDTcan be extrapolated from different scaling laws along withthe variation of thermal gradient 119866 in the fixed primaryspacing The solidliquid morphologies corresponding tothe different critical velocities are obtained by phase fieldsimulation Finally by comparing the simulation results withthe reference morphology the scaling law with differentexponents is evaluated Here 1198660=202 Kcm and 1198810=20120583ms are chosen as the referential control parameters Thecritical primary spacing for CDT is 1205821=160 120583m and thereferential morphology is shown in Figure 4(a) With 1205821=160
6 Advances in Condensed Matter Physics
(a)
10
10
100
(III)
(II)
(I)
(G 0 V0)
G (Kcm)V
(m
s)
=-13 =-13=-035=-021=-12 =-18
(b)
Figure 4 The critical microstructure of cell-dendritic transition at the benchmark (a) and the criteria of cell-dendrite transition accordingto the different power laws based on the benchmark (b)
V0=28ms V0=37ms V0=45ms V0=77msV0=28ms
Figure 5 The cellulardendritic morphologies with different pulling velocities when 119866=52 Kcm
120583m and 1198660=52 Kcm the different critical pulling velocitiesfor CDT are 119881I=77 120583ms 119881II=45 120583ms and 119881III=28 120583msrespectively for three groups of exponent parameters (-13-13) (-035 -021) and (-12 -18) as shown by the dot inFigure 4(b)
Figure 5 presents the cellulardendrite morphologies fordifferent pulling velocities with 1205821=160 120583m and1198660=52 KcmCompared with the referential morphology in Figure 4(a)the critical pulling velocity is around 35 120583ms The pullingvelocity 77 120583ms predicted by (-13 -13) is obviously largerthan the critical pulling velocity for CDT while the pullingvelocity 28 120583ms predicted by (-12 -18) is close to the criticalpulling velocity but with cellular morphology The pulling
velocity 45 120583ms from the new proposed scaling is also closeto the critical pulling velocity of CDT
The critical pulling velocity of CDT can be further foundwithin higher accuracy In the simplified form of the newproposed scaling law (formula (5)) the contribution of theterm 1205822119897119879 on the critical pulling velocity in (6) is overlookedHere by submitting 1205822=150Vminus059 into (6) the effect of theterm 1205822119897119879 can be revealed Considering the overall effectsof thermal gradient 119866 and pulling velocity 119881 the variationof the left-hand side of formula (5) with pulling velocity fortwo different thermal gradients is presented in Figure 6 Itshows that the exponent 120572 is near -035 with small thermalgradient 119866 However 120572 deviates from -035 gradually as the
Advances in Condensed Matter Physics 7
10 20 30 40 50 609000
12000
15000
18000
21000
24000
27000
G=52Kcm G=202Kcm
2minus1
4Ｆ ＄
12(Ｆ ４
)
V (ms)
Fitting by power law =-038Fitting by power law =-042
Figure 6 The variation of the value of right-hand side in (5) withpulling velocity for different thermal gradients
thermal gradient 119866 increases whichmeans that the influenceof pulling velocity 119881 on 1205822119897119879 has significant impact on119891(119897119879)with relatively small 119897119879 Considering the contribution ofpulling velocity 119881 on 119891(119897119879) Figure 6 gives the critical pullingvelocity of CDT as 37120583ms when 1205821=160 120583m and 1198660=52Kcm in the simulation system The interface morphologyfor 37 120583ms is very close to the reference morphology in thebenchmark
5 Results and Discussion
To summarize a simplified scaling law of CDT duringdirectional solidification is derived with considering thesidebranching dynamics The exponent parameters corre-sponding to the pulling velocity and thermal gradient inthe new scaling law are discussed The analysis shows thatthe exponent parameters in the scaling law vary with dif-ferent solidification systems and reconcile the discrepancyin previous experimental results The form of this scalinglaw can return back to the empirical one and reconciles thedifference of the exponent parameters in previous exper-iments The destabilizing mechanism of thermal gradientin the sidebranching dynamics can be revealed by lateraldiffusion length The new scaling law is also validated bya benchmark from quantitative phase field simulation Theappropriate experimental verification of the scaling law canbe similar to that done by Teng et al [9] With the apparatusone can use different systems and parameters to check thecellular-to-dendrite transition
Furthermore the proposed scaling law is more than theconciliation of the controversy in previous experiments Onone hand compared with previous scaling law the newscaling law is with more physical foundation related to thesidebranching dynamics On the other hand it indicates
that the thermal gradient and pulling velocity are coupledtogether in describing the CDT within a large range param-eter space Only in local parameter space the scaling law ofCDT has a simple form consisted with previous experimentalresults where the thermal gradient and pulling velocity aredecoupled Therefore the proposed scaling law is with moreprecision in predicting CDT in a large range parameter spacecompared with previous investigations
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The work was supported by the fund of the State KeyLaboratory of Solidification Processing in NWPU (Grant noSKLSP201725)
References
[1] J S Langer ldquoInstabilities and pattern formation in crystalgrowthrdquoReviews ofModern Physics vol 52 no 1 pp 1ndash28 1980
[2] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993
[3] W J Boettinger S R Coriell A L Greer et al ldquoSolidificationmicrostructures recent developments future directionsrdquo ActaMaterialia vol 48 no 1 pp 43ndash70 2000
[4] A Karma and W-J Rappel ldquoPhase-field model of dendriticsidebranchingwith thermal noiserdquoPhysical Review E StatisticalPhysics Plasmas Fluids and Related Interdisciplinary Topicsvol 60 no 4 pp 3614ndash3625 1999
[5] M EGlicksman J S Lowengrub SW Li et al ldquoAdeterministicmechanism for dendritic solidification kineticsrdquoThe Journal oftheMinerals Metals ampMaterials Society vol 59 no 8 pp 27ndash342007
[6] Z Wang J Wang and G Yang ldquoPhase-field investigationof effects of surface-tension anisotropy on deterministic side-branching in solutal dendritic growthrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 78 no 4Article ID 042601 2008
[7] M Georgelin and A Pocheau ldquoOnset of sidebranching indirectional solidificationrdquo Physical Review E Statistical PhysicsPlasmas Fluids and Related Interdisciplinary Topics vol 57 no3 pp 3189ndash3203 1998
[8] R Trivedi Y Shen and S Liu ldquoCellular-to-dendritic transitionduring the directional solidification of binary alloysrdquo Metal-lurgical and Materials Transactions A Physical Metallurgy andMaterials Science vol 34 no 2 pp 395ndash401 2003
[9] J Teng S Liu and R Trivedi ldquoOnset of sidewise instabilityand cell-dendrite transition in directional solidificationrdquo ActaMaterialia vol 57 no 12 pp 3497ndash3508 2009
[10] G L Ding On primary dendritic spacing during unidirectionalsolidification [PhD thesis] Northwestern Polytechnical Univer-sity Xirsquoan China 1997
[11] E Acer E Cadırlı H Erol H Kaya and M Gunduz ldquoEffectsof growth rates and compositions on dendrite arm spacings indirectionally solidified Al-Zn alloysrdquoMetallurgical and Materi-als Transactions A Physical Metallurgy and Materials Sciencevol 48 no 12 pp 5911ndash5923 2017
8 Advances in Condensed Matter Physics
[12] A Pocheau S Bodea and M Georgelin ldquoSelf-organized den-dritic sidebranching in directional solidification sidebranchcoherence within uncorrelated burstsrdquo Physical Review E Sta-tistical Nonlinear and SoftMatter Physics vol 80 no 3 ArticleID 031601 2009
[13] J S Kirkaldy L X Liu and A Kroupa ldquoThin film forcedvelocity cells and cellular dendrites-II Analysis of datardquo ActaMetallurgica et Materialia vol 43 no 8 pp 2905ndash2915 1995
[14] K Somboonsuk J T Mason and R Trivedi ldquoInterdendriticspacing part I Experimental studiesrdquo Metallurgical Transac-tions A Physical Metallurgy and Materials Science vol 15 no6 pp 967ndash975 1984
[15] Z JWang J CWang andGC Yang ldquoPhase field investigationon the selection of initial sidebranch spacing in directionalsolidificationrdquo IOP Conference Series Materials Science andEngineering vol 27 no 1 p 012009 2012
[16] B Echebarria A Karma and S Gurevich ldquoOnset of side-branching in directional solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 81 no 2Article ID 021608 2010
[17] Z Wang J Li J Wang and Y Zhou ldquoPhase field modelingthe selection mechanism of primary dendritic spacing indirectional solidificationrdquo Acta Materialia vol 60 no 5 pp1957ndash1964 2012
[18] B Echebarria R Folch A Karma and M Plapp ldquoQuantitativephase-field model of alloy solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 70 no 6Article ID 061604 2004
[19] B J Spencer andH EHuppert ldquoRelationship between dendritetip characteristics and dendrite spacings in alloy directionalsolidificationrdquo Journal of Crystal Growth vol 200 no 1-2 pp287ndash296 1999
[20] Z Wang J Wang and G Yang ldquoOnset of initial planarinstability with surface-tension anisotropy during directionalsolidificationrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 80 no 5 Article ID 052603 2009
[21] Z Wang J Wang and G Yang ldquoFourier synthesis predictingonset of the initial instability during directional solidificationrdquoApplied Physics Letters vol 94 no 6 p 061920 2009
[22] J J Xu Interfacial Wave Theory of Pattern Formation in Solidi-fication Dendrites Fingers Cells and Free Boundary SpringerSeries in Synergetics Springer International Publishing 2ndedition 2017
[23] G Agez M G Clerc E Louvergneaux and R G RojasldquoBifurcations of emerging patterns in the presence of additivenoiserdquo Physical Review E Statistical Nonlinear and Soft MatterPhysics vol 87 no 4 Article ID 042919 2013
[24] I Steinbach ldquoEffect of interface anisotropy on spacing selectionin constrained dendrite growthrdquoActaMaterialia vol 56 no 18pp 4965ndash4971 2008
[25] S Gurevich M Amoorezaei and N Provatas ldquoPhase-fieldstudy of spacing evolution during transient growthrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 82no 5 Article ID 051606 2010
[26] M Amoorezaei S Gurevich and N Provatas ldquoSpacing char-acterization in Al-Cu alloys directionally solidified under tran-sient growth conditionsrdquo Acta Materialia vol 58 no 18 pp6115ndash6124 2010
[27] W Losert B Q Shi and H Z Cummins ldquoEvolution ofdendritic patterns during alloy solidification onset of the initialinstabilityrdquo Proceedings of the National Acadamy of Sciences ofthe United States of America vol 95 no 2 pp 431ndash438 1998
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
High Energy PhysicsAdvances in
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
AstronomyAdvances in
Antennas andPropagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
International Journal of
Geophysics
Advances inOpticalTechnologies
Hindawiwwwhindawicom
Volume 2018
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
Advances in Condensed Matter Physics 5
length which are related to the undercooling and thermalgradient respectively In experiments all these parametersvary in different systems with different parameters Accord-ingly the validity of the scaling law can be well checked withdesigned experiments In this research the new scaling lawwill be validated by a benchmark fromquantitative phase fieldsimulation instead
Note that the noise effects on the sidebranching dynamicsin dendrite growth have been a controversial issue formany years [22] However there is an agreement that thepresence of stochastic noise will not affect the frequency ofsidebranching while the value of the amplitude is a functionof noise intensity level Just as predicted by [23] the transitionof cellular to dendrite growth may be modified by the noiseintensity level In the derivation of the scaling law thesidebranching amplitude is considered as a function of threelength scales that are independent of the noise Thereforethe noise may affect the transition points but the scaling lawbetween these lengths still exists At an adequate noise levelsimilar to that in the experiment the scaling law is valid
4 Validation of the Scaling Law byPhase Field Simulation
The development of phase field method makes it possible toquantitatively investigate microstructure evolution in solidi-fication [18] and the quantitative phase field simulation hasbeen widely used to investigate the sidebranching dynamicsin crystal growth [6 15 16] and the primary spacing selec-tion mechanism in directional solidification [17 24ndash26] Tofurther validate the scaling law the quantitative phase fieldmethod [18] is employed For the directional solidificationsimulation by (7) and (8) assume one-sided diffusion andfrozen temperature approximation in which119879 = 119879119862+119866119911 and119879119862 is the temperature at the cooling end and 119866 is the thermalgradient along 119911-axis The dynamic evolution equations ofphase field model in the moving frame with pulling velocity119881 are
1205910 (1 minus (1 minus 119896) 119911 + (119898119888infin119896) 119866119897119879 ) (120597119905120601 minus 119881120597119911120601)= nabla (119882(120579)2 nabla120601) minus 120597119909 [119882 (120579)1198821015840 (120579) 120597119910120601]+ 120597119910 [119882 (120579)1198821015840 (120579) 120597119909120601] + 120601 minus 1206013 + 120582 (1 minus 1206012)2 (119880
+ 119911 + (119898119888infin119896) 119866119897119879 )
(7)
120597119905119888 minus 119881120597119911119888 = nabla sdot [119863119871 1 minus 1206011 + 119896 minus (1 minus 119896) 120601nabla119888
+ (119863119871 1 minus 1206011 + 119896 minus (1 minus 119896) 120601 + 1198820radic2
(120597119905120601 minus 119881120597119911120601)1003816100381610038161003816nabla1206011003816100381610038161003816 )
sdot 119888 (1 minus 119896)1 + 119896 minus (1 minus 119896) 120601nabla120601]
(8)
with
119882(120579) = 1198820 (1 + 1205744 cos 4120579) (9)
119880 = ((2119896119888119888infin) (1 + 119896 minus (1 minus 119896) 120601) minus 1)(1 minus 119896) (10)
119897119879 = |119898| 119888infin (1 minus 119896)(119896119866) (11)
where 1198820 are the parameters of the interface thickness 1205910is the relaxation time for phase field model and 120582 is thecoupling constant which are related to physical quantitiesby 1198890 = 11988611198820120582 and 1205910 = 119886212058211988220 119863 1205744 is the anisotropicintensity of the surface tension 120579 is the angle between thenormal vector of the interface and the preferred orientation119898 is the liquidus slope k is the partition coefficient and119888infin is the concentration in the far away field Here 1198890 =Γ(119898119888infin(1-119896)119896) 1198861=08839 and 1198862=06267 and Γ is theGibbs-Thomson coefficient
The transparent alloy SCN-043wtC152 is adoptedwhich has been widely used to investigate the evolution ofdendritic pattern [27] The chemical diffusion coefficient ofthe liquid phase is 119863 asymp 045times10minus9 m2s partition coefficient119896=005 Gibbs-Thomson coefficient Γ = 648times10minus8 Km andthe slope of liquidus line 119898=-542 Kmol The surface tensionanisotropy intensity is assumed as 1205744=0005
This quantitative phase field simulation on dendriticgrowth with the presented parameters has been widely per-formed in our previous investigations [15 17]The phase fieldsimulation on the sidebranching indicates that 1205822=150Vminus059and the parameters 119886 and 119899 in (6) are about 04 and 5respectively Then the exponents in the new proposed scalinglaw for the simulation system are 120572 = minus035 and 120573 = minus021according to formula (5) which are different from 120572 = 120573 =minus13 [8 9] and 120572 = minus12 and 120573 = minus18 [7] These threedifferent groups of exponents will be quantitatively examinedaccording to the phase field simulation results
Here a benchmark is designed to directly compare theexponents in different scaling lawsThe exponents are usuallyobtained by fitting plenty of simulation or experiment resultswith different control parameters However heavy workloadand artificial judgment on the onset of sidebranching arerequired by this method In a different way we design abenchmark instead of finding the exponents in the simulationsystem as follows The steady state of interface morphologyin specific primary spacing with onset of sidebranchingis firstly presented where the thermal gradient 1198660 andpulling velocity 1198810 as well as the morphology will be thereferences Then the critical pulling velocity for the CDTcan be extrapolated from different scaling laws along withthe variation of thermal gradient 119866 in the fixed primaryspacing The solidliquid morphologies corresponding tothe different critical velocities are obtained by phase fieldsimulation Finally by comparing the simulation results withthe reference morphology the scaling law with differentexponents is evaluated Here 1198660=202 Kcm and 1198810=20120583ms are chosen as the referential control parameters Thecritical primary spacing for CDT is 1205821=160 120583m and thereferential morphology is shown in Figure 4(a) With 1205821=160
6 Advances in Condensed Matter Physics
(a)
10
10
100
(III)
(II)
(I)
(G 0 V0)
G (Kcm)V
(m
s)
=-13 =-13=-035=-021=-12 =-18
(b)
Figure 4 The critical microstructure of cell-dendritic transition at the benchmark (a) and the criteria of cell-dendrite transition accordingto the different power laws based on the benchmark (b)
V0=28ms V0=37ms V0=45ms V0=77msV0=28ms
Figure 5 The cellulardendritic morphologies with different pulling velocities when 119866=52 Kcm
120583m and 1198660=52 Kcm the different critical pulling velocitiesfor CDT are 119881I=77 120583ms 119881II=45 120583ms and 119881III=28 120583msrespectively for three groups of exponent parameters (-13-13) (-035 -021) and (-12 -18) as shown by the dot inFigure 4(b)
Figure 5 presents the cellulardendrite morphologies fordifferent pulling velocities with 1205821=160 120583m and1198660=52 KcmCompared with the referential morphology in Figure 4(a)the critical pulling velocity is around 35 120583ms The pullingvelocity 77 120583ms predicted by (-13 -13) is obviously largerthan the critical pulling velocity for CDT while the pullingvelocity 28 120583ms predicted by (-12 -18) is close to the criticalpulling velocity but with cellular morphology The pulling
velocity 45 120583ms from the new proposed scaling is also closeto the critical pulling velocity of CDT
The critical pulling velocity of CDT can be further foundwithin higher accuracy In the simplified form of the newproposed scaling law (formula (5)) the contribution of theterm 1205822119897119879 on the critical pulling velocity in (6) is overlookedHere by submitting 1205822=150Vminus059 into (6) the effect of theterm 1205822119897119879 can be revealed Considering the overall effectsof thermal gradient 119866 and pulling velocity 119881 the variationof the left-hand side of formula (5) with pulling velocity fortwo different thermal gradients is presented in Figure 6 Itshows that the exponent 120572 is near -035 with small thermalgradient 119866 However 120572 deviates from -035 gradually as the
Advances in Condensed Matter Physics 7
10 20 30 40 50 609000
12000
15000
18000
21000
24000
27000
G=52Kcm G=202Kcm
2minus1
4Ｆ ＄
12(Ｆ ４
)
V (ms)
Fitting by power law =-038Fitting by power law =-042
Figure 6 The variation of the value of right-hand side in (5) withpulling velocity for different thermal gradients
thermal gradient 119866 increases whichmeans that the influenceof pulling velocity 119881 on 1205822119897119879 has significant impact on119891(119897119879)with relatively small 119897119879 Considering the contribution ofpulling velocity 119881 on 119891(119897119879) Figure 6 gives the critical pullingvelocity of CDT as 37120583ms when 1205821=160 120583m and 1198660=52Kcm in the simulation system The interface morphologyfor 37 120583ms is very close to the reference morphology in thebenchmark
5 Results and Discussion
To summarize a simplified scaling law of CDT duringdirectional solidification is derived with considering thesidebranching dynamics The exponent parameters corre-sponding to the pulling velocity and thermal gradient inthe new scaling law are discussed The analysis shows thatthe exponent parameters in the scaling law vary with dif-ferent solidification systems and reconcile the discrepancyin previous experimental results The form of this scalinglaw can return back to the empirical one and reconciles thedifference of the exponent parameters in previous exper-iments The destabilizing mechanism of thermal gradientin the sidebranching dynamics can be revealed by lateraldiffusion length The new scaling law is also validated bya benchmark from quantitative phase field simulation Theappropriate experimental verification of the scaling law canbe similar to that done by Teng et al [9] With the apparatusone can use different systems and parameters to check thecellular-to-dendrite transition
Furthermore the proposed scaling law is more than theconciliation of the controversy in previous experiments Onone hand compared with previous scaling law the newscaling law is with more physical foundation related to thesidebranching dynamics On the other hand it indicates
that the thermal gradient and pulling velocity are coupledtogether in describing the CDT within a large range param-eter space Only in local parameter space the scaling law ofCDT has a simple form consisted with previous experimentalresults where the thermal gradient and pulling velocity aredecoupled Therefore the proposed scaling law is with moreprecision in predicting CDT in a large range parameter spacecompared with previous investigations
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The work was supported by the fund of the State KeyLaboratory of Solidification Processing in NWPU (Grant noSKLSP201725)
References
[1] J S Langer ldquoInstabilities and pattern formation in crystalgrowthrdquoReviews ofModern Physics vol 52 no 1 pp 1ndash28 1980
[2] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993
[3] W J Boettinger S R Coriell A L Greer et al ldquoSolidificationmicrostructures recent developments future directionsrdquo ActaMaterialia vol 48 no 1 pp 43ndash70 2000
[4] A Karma and W-J Rappel ldquoPhase-field model of dendriticsidebranchingwith thermal noiserdquoPhysical Review E StatisticalPhysics Plasmas Fluids and Related Interdisciplinary Topicsvol 60 no 4 pp 3614ndash3625 1999
[5] M EGlicksman J S Lowengrub SW Li et al ldquoAdeterministicmechanism for dendritic solidification kineticsrdquoThe Journal oftheMinerals Metals ampMaterials Society vol 59 no 8 pp 27ndash342007
[6] Z Wang J Wang and G Yang ldquoPhase-field investigationof effects of surface-tension anisotropy on deterministic side-branching in solutal dendritic growthrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 78 no 4Article ID 042601 2008
[7] M Georgelin and A Pocheau ldquoOnset of sidebranching indirectional solidificationrdquo Physical Review E Statistical PhysicsPlasmas Fluids and Related Interdisciplinary Topics vol 57 no3 pp 3189ndash3203 1998
[8] R Trivedi Y Shen and S Liu ldquoCellular-to-dendritic transitionduring the directional solidification of binary alloysrdquo Metal-lurgical and Materials Transactions A Physical Metallurgy andMaterials Science vol 34 no 2 pp 395ndash401 2003
[9] J Teng S Liu and R Trivedi ldquoOnset of sidewise instabilityand cell-dendrite transition in directional solidificationrdquo ActaMaterialia vol 57 no 12 pp 3497ndash3508 2009
[10] G L Ding On primary dendritic spacing during unidirectionalsolidification [PhD thesis] Northwestern Polytechnical Univer-sity Xirsquoan China 1997
[11] E Acer E Cadırlı H Erol H Kaya and M Gunduz ldquoEffectsof growth rates and compositions on dendrite arm spacings indirectionally solidified Al-Zn alloysrdquoMetallurgical and Materi-als Transactions A Physical Metallurgy and Materials Sciencevol 48 no 12 pp 5911ndash5923 2017
8 Advances in Condensed Matter Physics
[12] A Pocheau S Bodea and M Georgelin ldquoSelf-organized den-dritic sidebranching in directional solidification sidebranchcoherence within uncorrelated burstsrdquo Physical Review E Sta-tistical Nonlinear and SoftMatter Physics vol 80 no 3 ArticleID 031601 2009
[13] J S Kirkaldy L X Liu and A Kroupa ldquoThin film forcedvelocity cells and cellular dendrites-II Analysis of datardquo ActaMetallurgica et Materialia vol 43 no 8 pp 2905ndash2915 1995
[14] K Somboonsuk J T Mason and R Trivedi ldquoInterdendriticspacing part I Experimental studiesrdquo Metallurgical Transac-tions A Physical Metallurgy and Materials Science vol 15 no6 pp 967ndash975 1984
[15] Z JWang J CWang andGC Yang ldquoPhase field investigationon the selection of initial sidebranch spacing in directionalsolidificationrdquo IOP Conference Series Materials Science andEngineering vol 27 no 1 p 012009 2012
[16] B Echebarria A Karma and S Gurevich ldquoOnset of side-branching in directional solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 81 no 2Article ID 021608 2010
[17] Z Wang J Li J Wang and Y Zhou ldquoPhase field modelingthe selection mechanism of primary dendritic spacing indirectional solidificationrdquo Acta Materialia vol 60 no 5 pp1957ndash1964 2012
[18] B Echebarria R Folch A Karma and M Plapp ldquoQuantitativephase-field model of alloy solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 70 no 6Article ID 061604 2004
[19] B J Spencer andH EHuppert ldquoRelationship between dendritetip characteristics and dendrite spacings in alloy directionalsolidificationrdquo Journal of Crystal Growth vol 200 no 1-2 pp287ndash296 1999
[20] Z Wang J Wang and G Yang ldquoOnset of initial planarinstability with surface-tension anisotropy during directionalsolidificationrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 80 no 5 Article ID 052603 2009
[21] Z Wang J Wang and G Yang ldquoFourier synthesis predictingonset of the initial instability during directional solidificationrdquoApplied Physics Letters vol 94 no 6 p 061920 2009
[22] J J Xu Interfacial Wave Theory of Pattern Formation in Solidi-fication Dendrites Fingers Cells and Free Boundary SpringerSeries in Synergetics Springer International Publishing 2ndedition 2017
[23] G Agez M G Clerc E Louvergneaux and R G RojasldquoBifurcations of emerging patterns in the presence of additivenoiserdquo Physical Review E Statistical Nonlinear and Soft MatterPhysics vol 87 no 4 Article ID 042919 2013
[24] I Steinbach ldquoEffect of interface anisotropy on spacing selectionin constrained dendrite growthrdquoActaMaterialia vol 56 no 18pp 4965ndash4971 2008
[25] S Gurevich M Amoorezaei and N Provatas ldquoPhase-fieldstudy of spacing evolution during transient growthrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 82no 5 Article ID 051606 2010
[26] M Amoorezaei S Gurevich and N Provatas ldquoSpacing char-acterization in Al-Cu alloys directionally solidified under tran-sient growth conditionsrdquo Acta Materialia vol 58 no 18 pp6115ndash6124 2010
[27] W Losert B Q Shi and H Z Cummins ldquoEvolution ofdendritic patterns during alloy solidification onset of the initialinstabilityrdquo Proceedings of the National Acadamy of Sciences ofthe United States of America vol 95 no 2 pp 431ndash438 1998
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
High Energy PhysicsAdvances in
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
AstronomyAdvances in
Antennas andPropagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
International Journal of
Geophysics
Advances inOpticalTechnologies
Hindawiwwwhindawicom
Volume 2018
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
6 Advances in Condensed Matter Physics
(a)
10
10
100
(III)
(II)
(I)
(G 0 V0)
G (Kcm)V
(m
s)
=-13 =-13=-035=-021=-12 =-18
(b)
Figure 4 The critical microstructure of cell-dendritic transition at the benchmark (a) and the criteria of cell-dendrite transition accordingto the different power laws based on the benchmark (b)
V0=28ms V0=37ms V0=45ms V0=77msV0=28ms
Figure 5 The cellulardendritic morphologies with different pulling velocities when 119866=52 Kcm
120583m and 1198660=52 Kcm the different critical pulling velocitiesfor CDT are 119881I=77 120583ms 119881II=45 120583ms and 119881III=28 120583msrespectively for three groups of exponent parameters (-13-13) (-035 -021) and (-12 -18) as shown by the dot inFigure 4(b)
Figure 5 presents the cellulardendrite morphologies fordifferent pulling velocities with 1205821=160 120583m and1198660=52 KcmCompared with the referential morphology in Figure 4(a)the critical pulling velocity is around 35 120583ms The pullingvelocity 77 120583ms predicted by (-13 -13) is obviously largerthan the critical pulling velocity for CDT while the pullingvelocity 28 120583ms predicted by (-12 -18) is close to the criticalpulling velocity but with cellular morphology The pulling
velocity 45 120583ms from the new proposed scaling is also closeto the critical pulling velocity of CDT
The critical pulling velocity of CDT can be further foundwithin higher accuracy In the simplified form of the newproposed scaling law (formula (5)) the contribution of theterm 1205822119897119879 on the critical pulling velocity in (6) is overlookedHere by submitting 1205822=150Vminus059 into (6) the effect of theterm 1205822119897119879 can be revealed Considering the overall effectsof thermal gradient 119866 and pulling velocity 119881 the variationof the left-hand side of formula (5) with pulling velocity fortwo different thermal gradients is presented in Figure 6 Itshows that the exponent 120572 is near -035 with small thermalgradient 119866 However 120572 deviates from -035 gradually as the
Advances in Condensed Matter Physics 7
10 20 30 40 50 609000
12000
15000
18000
21000
24000
27000
G=52Kcm G=202Kcm
2minus1
4Ｆ ＄
12(Ｆ ４
)
V (ms)
Fitting by power law =-038Fitting by power law =-042
Figure 6 The variation of the value of right-hand side in (5) withpulling velocity for different thermal gradients
thermal gradient 119866 increases whichmeans that the influenceof pulling velocity 119881 on 1205822119897119879 has significant impact on119891(119897119879)with relatively small 119897119879 Considering the contribution ofpulling velocity 119881 on 119891(119897119879) Figure 6 gives the critical pullingvelocity of CDT as 37120583ms when 1205821=160 120583m and 1198660=52Kcm in the simulation system The interface morphologyfor 37 120583ms is very close to the reference morphology in thebenchmark
5 Results and Discussion
To summarize a simplified scaling law of CDT duringdirectional solidification is derived with considering thesidebranching dynamics The exponent parameters corre-sponding to the pulling velocity and thermal gradient inthe new scaling law are discussed The analysis shows thatthe exponent parameters in the scaling law vary with dif-ferent solidification systems and reconcile the discrepancyin previous experimental results The form of this scalinglaw can return back to the empirical one and reconciles thedifference of the exponent parameters in previous exper-iments The destabilizing mechanism of thermal gradientin the sidebranching dynamics can be revealed by lateraldiffusion length The new scaling law is also validated bya benchmark from quantitative phase field simulation Theappropriate experimental verification of the scaling law canbe similar to that done by Teng et al [9] With the apparatusone can use different systems and parameters to check thecellular-to-dendrite transition
Furthermore the proposed scaling law is more than theconciliation of the controversy in previous experiments Onone hand compared with previous scaling law the newscaling law is with more physical foundation related to thesidebranching dynamics On the other hand it indicates
that the thermal gradient and pulling velocity are coupledtogether in describing the CDT within a large range param-eter space Only in local parameter space the scaling law ofCDT has a simple form consisted with previous experimentalresults where the thermal gradient and pulling velocity aredecoupled Therefore the proposed scaling law is with moreprecision in predicting CDT in a large range parameter spacecompared with previous investigations
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The work was supported by the fund of the State KeyLaboratory of Solidification Processing in NWPU (Grant noSKLSP201725)
References
[1] J S Langer ldquoInstabilities and pattern formation in crystalgrowthrdquoReviews ofModern Physics vol 52 no 1 pp 1ndash28 1980
[2] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993
[3] W J Boettinger S R Coriell A L Greer et al ldquoSolidificationmicrostructures recent developments future directionsrdquo ActaMaterialia vol 48 no 1 pp 43ndash70 2000
[4] A Karma and W-J Rappel ldquoPhase-field model of dendriticsidebranchingwith thermal noiserdquoPhysical Review E StatisticalPhysics Plasmas Fluids and Related Interdisciplinary Topicsvol 60 no 4 pp 3614ndash3625 1999
[5] M EGlicksman J S Lowengrub SW Li et al ldquoAdeterministicmechanism for dendritic solidification kineticsrdquoThe Journal oftheMinerals Metals ampMaterials Society vol 59 no 8 pp 27ndash342007
[6] Z Wang J Wang and G Yang ldquoPhase-field investigationof effects of surface-tension anisotropy on deterministic side-branching in solutal dendritic growthrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 78 no 4Article ID 042601 2008
[7] M Georgelin and A Pocheau ldquoOnset of sidebranching indirectional solidificationrdquo Physical Review E Statistical PhysicsPlasmas Fluids and Related Interdisciplinary Topics vol 57 no3 pp 3189ndash3203 1998
[8] R Trivedi Y Shen and S Liu ldquoCellular-to-dendritic transitionduring the directional solidification of binary alloysrdquo Metal-lurgical and Materials Transactions A Physical Metallurgy andMaterials Science vol 34 no 2 pp 395ndash401 2003
[9] J Teng S Liu and R Trivedi ldquoOnset of sidewise instabilityand cell-dendrite transition in directional solidificationrdquo ActaMaterialia vol 57 no 12 pp 3497ndash3508 2009
[10] G L Ding On primary dendritic spacing during unidirectionalsolidification [PhD thesis] Northwestern Polytechnical Univer-sity Xirsquoan China 1997
[11] E Acer E Cadırlı H Erol H Kaya and M Gunduz ldquoEffectsof growth rates and compositions on dendrite arm spacings indirectionally solidified Al-Zn alloysrdquoMetallurgical and Materi-als Transactions A Physical Metallurgy and Materials Sciencevol 48 no 12 pp 5911ndash5923 2017
8 Advances in Condensed Matter Physics
[12] A Pocheau S Bodea and M Georgelin ldquoSelf-organized den-dritic sidebranching in directional solidification sidebranchcoherence within uncorrelated burstsrdquo Physical Review E Sta-tistical Nonlinear and SoftMatter Physics vol 80 no 3 ArticleID 031601 2009
[13] J S Kirkaldy L X Liu and A Kroupa ldquoThin film forcedvelocity cells and cellular dendrites-II Analysis of datardquo ActaMetallurgica et Materialia vol 43 no 8 pp 2905ndash2915 1995
[14] K Somboonsuk J T Mason and R Trivedi ldquoInterdendriticspacing part I Experimental studiesrdquo Metallurgical Transac-tions A Physical Metallurgy and Materials Science vol 15 no6 pp 967ndash975 1984
[15] Z JWang J CWang andGC Yang ldquoPhase field investigationon the selection of initial sidebranch spacing in directionalsolidificationrdquo IOP Conference Series Materials Science andEngineering vol 27 no 1 p 012009 2012
[16] B Echebarria A Karma and S Gurevich ldquoOnset of side-branching in directional solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 81 no 2Article ID 021608 2010
[17] Z Wang J Li J Wang and Y Zhou ldquoPhase field modelingthe selection mechanism of primary dendritic spacing indirectional solidificationrdquo Acta Materialia vol 60 no 5 pp1957ndash1964 2012
[18] B Echebarria R Folch A Karma and M Plapp ldquoQuantitativephase-field model of alloy solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 70 no 6Article ID 061604 2004
[19] B J Spencer andH EHuppert ldquoRelationship between dendritetip characteristics and dendrite spacings in alloy directionalsolidificationrdquo Journal of Crystal Growth vol 200 no 1-2 pp287ndash296 1999
[20] Z Wang J Wang and G Yang ldquoOnset of initial planarinstability with surface-tension anisotropy during directionalsolidificationrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 80 no 5 Article ID 052603 2009
[21] Z Wang J Wang and G Yang ldquoFourier synthesis predictingonset of the initial instability during directional solidificationrdquoApplied Physics Letters vol 94 no 6 p 061920 2009
[22] J J Xu Interfacial Wave Theory of Pattern Formation in Solidi-fication Dendrites Fingers Cells and Free Boundary SpringerSeries in Synergetics Springer International Publishing 2ndedition 2017
[23] G Agez M G Clerc E Louvergneaux and R G RojasldquoBifurcations of emerging patterns in the presence of additivenoiserdquo Physical Review E Statistical Nonlinear and Soft MatterPhysics vol 87 no 4 Article ID 042919 2013
[24] I Steinbach ldquoEffect of interface anisotropy on spacing selectionin constrained dendrite growthrdquoActaMaterialia vol 56 no 18pp 4965ndash4971 2008
[25] S Gurevich M Amoorezaei and N Provatas ldquoPhase-fieldstudy of spacing evolution during transient growthrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 82no 5 Article ID 051606 2010
[26] M Amoorezaei S Gurevich and N Provatas ldquoSpacing char-acterization in Al-Cu alloys directionally solidified under tran-sient growth conditionsrdquo Acta Materialia vol 58 no 18 pp6115ndash6124 2010
[27] W Losert B Q Shi and H Z Cummins ldquoEvolution ofdendritic patterns during alloy solidification onset of the initialinstabilityrdquo Proceedings of the National Acadamy of Sciences ofthe United States of America vol 95 no 2 pp 431ndash438 1998
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
High Energy PhysicsAdvances in
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
AstronomyAdvances in
Antennas andPropagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
International Journal of
Geophysics
Advances inOpticalTechnologies
Hindawiwwwhindawicom
Volume 2018
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
Advances in Condensed Matter Physics 7
10 20 30 40 50 609000
12000
15000
18000
21000
24000
27000
G=52Kcm G=202Kcm
2minus1
4Ｆ ＄
12(Ｆ ４
)
V (ms)
Fitting by power law =-038Fitting by power law =-042
Figure 6 The variation of the value of right-hand side in (5) withpulling velocity for different thermal gradients
thermal gradient 119866 increases whichmeans that the influenceof pulling velocity 119881 on 1205822119897119879 has significant impact on119891(119897119879)with relatively small 119897119879 Considering the contribution ofpulling velocity 119881 on 119891(119897119879) Figure 6 gives the critical pullingvelocity of CDT as 37120583ms when 1205821=160 120583m and 1198660=52Kcm in the simulation system The interface morphologyfor 37 120583ms is very close to the reference morphology in thebenchmark
5 Results and Discussion
To summarize a simplified scaling law of CDT duringdirectional solidification is derived with considering thesidebranching dynamics The exponent parameters corre-sponding to the pulling velocity and thermal gradient inthe new scaling law are discussed The analysis shows thatthe exponent parameters in the scaling law vary with dif-ferent solidification systems and reconcile the discrepancyin previous experimental results The form of this scalinglaw can return back to the empirical one and reconciles thedifference of the exponent parameters in previous exper-iments The destabilizing mechanism of thermal gradientin the sidebranching dynamics can be revealed by lateraldiffusion length The new scaling law is also validated bya benchmark from quantitative phase field simulation Theappropriate experimental verification of the scaling law canbe similar to that done by Teng et al [9] With the apparatusone can use different systems and parameters to check thecellular-to-dendrite transition
Furthermore the proposed scaling law is more than theconciliation of the controversy in previous experiments Onone hand compared with previous scaling law the newscaling law is with more physical foundation related to thesidebranching dynamics On the other hand it indicates
that the thermal gradient and pulling velocity are coupledtogether in describing the CDT within a large range param-eter space Only in local parameter space the scaling law ofCDT has a simple form consisted with previous experimentalresults where the thermal gradient and pulling velocity aredecoupled Therefore the proposed scaling law is with moreprecision in predicting CDT in a large range parameter spacecompared with previous investigations
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The work was supported by the fund of the State KeyLaboratory of Solidification Processing in NWPU (Grant noSKLSP201725)
References
[1] J S Langer ldquoInstabilities and pattern formation in crystalgrowthrdquoReviews ofModern Physics vol 52 no 1 pp 1ndash28 1980
[2] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993
[3] W J Boettinger S R Coriell A L Greer et al ldquoSolidificationmicrostructures recent developments future directionsrdquo ActaMaterialia vol 48 no 1 pp 43ndash70 2000
[4] A Karma and W-J Rappel ldquoPhase-field model of dendriticsidebranchingwith thermal noiserdquoPhysical Review E StatisticalPhysics Plasmas Fluids and Related Interdisciplinary Topicsvol 60 no 4 pp 3614ndash3625 1999
[5] M EGlicksman J S Lowengrub SW Li et al ldquoAdeterministicmechanism for dendritic solidification kineticsrdquoThe Journal oftheMinerals Metals ampMaterials Society vol 59 no 8 pp 27ndash342007
[6] Z Wang J Wang and G Yang ldquoPhase-field investigationof effects of surface-tension anisotropy on deterministic side-branching in solutal dendritic growthrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 78 no 4Article ID 042601 2008
[7] M Georgelin and A Pocheau ldquoOnset of sidebranching indirectional solidificationrdquo Physical Review E Statistical PhysicsPlasmas Fluids and Related Interdisciplinary Topics vol 57 no3 pp 3189ndash3203 1998
[8] R Trivedi Y Shen and S Liu ldquoCellular-to-dendritic transitionduring the directional solidification of binary alloysrdquo Metal-lurgical and Materials Transactions A Physical Metallurgy andMaterials Science vol 34 no 2 pp 395ndash401 2003
[9] J Teng S Liu and R Trivedi ldquoOnset of sidewise instabilityand cell-dendrite transition in directional solidificationrdquo ActaMaterialia vol 57 no 12 pp 3497ndash3508 2009
[10] G L Ding On primary dendritic spacing during unidirectionalsolidification [PhD thesis] Northwestern Polytechnical Univer-sity Xirsquoan China 1997
[11] E Acer E Cadırlı H Erol H Kaya and M Gunduz ldquoEffectsof growth rates and compositions on dendrite arm spacings indirectionally solidified Al-Zn alloysrdquoMetallurgical and Materi-als Transactions A Physical Metallurgy and Materials Sciencevol 48 no 12 pp 5911ndash5923 2017
8 Advances in Condensed Matter Physics
[12] A Pocheau S Bodea and M Georgelin ldquoSelf-organized den-dritic sidebranching in directional solidification sidebranchcoherence within uncorrelated burstsrdquo Physical Review E Sta-tistical Nonlinear and SoftMatter Physics vol 80 no 3 ArticleID 031601 2009
[13] J S Kirkaldy L X Liu and A Kroupa ldquoThin film forcedvelocity cells and cellular dendrites-II Analysis of datardquo ActaMetallurgica et Materialia vol 43 no 8 pp 2905ndash2915 1995
[14] K Somboonsuk J T Mason and R Trivedi ldquoInterdendriticspacing part I Experimental studiesrdquo Metallurgical Transac-tions A Physical Metallurgy and Materials Science vol 15 no6 pp 967ndash975 1984
[15] Z JWang J CWang andGC Yang ldquoPhase field investigationon the selection of initial sidebranch spacing in directionalsolidificationrdquo IOP Conference Series Materials Science andEngineering vol 27 no 1 p 012009 2012
[16] B Echebarria A Karma and S Gurevich ldquoOnset of side-branching in directional solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 81 no 2Article ID 021608 2010
[17] Z Wang J Li J Wang and Y Zhou ldquoPhase field modelingthe selection mechanism of primary dendritic spacing indirectional solidificationrdquo Acta Materialia vol 60 no 5 pp1957ndash1964 2012
[18] B Echebarria R Folch A Karma and M Plapp ldquoQuantitativephase-field model of alloy solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 70 no 6Article ID 061604 2004
[19] B J Spencer andH EHuppert ldquoRelationship between dendritetip characteristics and dendrite spacings in alloy directionalsolidificationrdquo Journal of Crystal Growth vol 200 no 1-2 pp287ndash296 1999
[20] Z Wang J Wang and G Yang ldquoOnset of initial planarinstability with surface-tension anisotropy during directionalsolidificationrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 80 no 5 Article ID 052603 2009
[21] Z Wang J Wang and G Yang ldquoFourier synthesis predictingonset of the initial instability during directional solidificationrdquoApplied Physics Letters vol 94 no 6 p 061920 2009
[22] J J Xu Interfacial Wave Theory of Pattern Formation in Solidi-fication Dendrites Fingers Cells and Free Boundary SpringerSeries in Synergetics Springer International Publishing 2ndedition 2017
[23] G Agez M G Clerc E Louvergneaux and R G RojasldquoBifurcations of emerging patterns in the presence of additivenoiserdquo Physical Review E Statistical Nonlinear and Soft MatterPhysics vol 87 no 4 Article ID 042919 2013
[24] I Steinbach ldquoEffect of interface anisotropy on spacing selectionin constrained dendrite growthrdquoActaMaterialia vol 56 no 18pp 4965ndash4971 2008
[25] S Gurevich M Amoorezaei and N Provatas ldquoPhase-fieldstudy of spacing evolution during transient growthrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 82no 5 Article ID 051606 2010
[26] M Amoorezaei S Gurevich and N Provatas ldquoSpacing char-acterization in Al-Cu alloys directionally solidified under tran-sient growth conditionsrdquo Acta Materialia vol 58 no 18 pp6115ndash6124 2010
[27] W Losert B Q Shi and H Z Cummins ldquoEvolution ofdendritic patterns during alloy solidification onset of the initialinstabilityrdquo Proceedings of the National Acadamy of Sciences ofthe United States of America vol 95 no 2 pp 431ndash438 1998
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
High Energy PhysicsAdvances in
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
AstronomyAdvances in
Antennas andPropagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
International Journal of
Geophysics
Advances inOpticalTechnologies
Hindawiwwwhindawicom
Volume 2018
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
8 Advances in Condensed Matter Physics
[12] A Pocheau S Bodea and M Georgelin ldquoSelf-organized den-dritic sidebranching in directional solidification sidebranchcoherence within uncorrelated burstsrdquo Physical Review E Sta-tistical Nonlinear and SoftMatter Physics vol 80 no 3 ArticleID 031601 2009
[13] J S Kirkaldy L X Liu and A Kroupa ldquoThin film forcedvelocity cells and cellular dendrites-II Analysis of datardquo ActaMetallurgica et Materialia vol 43 no 8 pp 2905ndash2915 1995
[14] K Somboonsuk J T Mason and R Trivedi ldquoInterdendriticspacing part I Experimental studiesrdquo Metallurgical Transac-tions A Physical Metallurgy and Materials Science vol 15 no6 pp 967ndash975 1984
[15] Z JWang J CWang andGC Yang ldquoPhase field investigationon the selection of initial sidebranch spacing in directionalsolidificationrdquo IOP Conference Series Materials Science andEngineering vol 27 no 1 p 012009 2012
[16] B Echebarria A Karma and S Gurevich ldquoOnset of side-branching in directional solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 81 no 2Article ID 021608 2010
[17] Z Wang J Li J Wang and Y Zhou ldquoPhase field modelingthe selection mechanism of primary dendritic spacing indirectional solidificationrdquo Acta Materialia vol 60 no 5 pp1957ndash1964 2012
[18] B Echebarria R Folch A Karma and M Plapp ldquoQuantitativephase-field model of alloy solidificationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 70 no 6Article ID 061604 2004
[19] B J Spencer andH EHuppert ldquoRelationship between dendritetip characteristics and dendrite spacings in alloy directionalsolidificationrdquo Journal of Crystal Growth vol 200 no 1-2 pp287ndash296 1999
[20] Z Wang J Wang and G Yang ldquoOnset of initial planarinstability with surface-tension anisotropy during directionalsolidificationrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 80 no 5 Article ID 052603 2009
[21] Z Wang J Wang and G Yang ldquoFourier synthesis predictingonset of the initial instability during directional solidificationrdquoApplied Physics Letters vol 94 no 6 p 061920 2009
[22] J J Xu Interfacial Wave Theory of Pattern Formation in Solidi-fication Dendrites Fingers Cells and Free Boundary SpringerSeries in Synergetics Springer International Publishing 2ndedition 2017
[23] G Agez M G Clerc E Louvergneaux and R G RojasldquoBifurcations of emerging patterns in the presence of additivenoiserdquo Physical Review E Statistical Nonlinear and Soft MatterPhysics vol 87 no 4 Article ID 042919 2013
[24] I Steinbach ldquoEffect of interface anisotropy on spacing selectionin constrained dendrite growthrdquoActaMaterialia vol 56 no 18pp 4965ndash4971 2008
[25] S Gurevich M Amoorezaei and N Provatas ldquoPhase-fieldstudy of spacing evolution during transient growthrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 82no 5 Article ID 051606 2010
[26] M Amoorezaei S Gurevich and N Provatas ldquoSpacing char-acterization in Al-Cu alloys directionally solidified under tran-sient growth conditionsrdquo Acta Materialia vol 58 no 18 pp6115ndash6124 2010
[27] W Losert B Q Shi and H Z Cummins ldquoEvolution ofdendritic patterns during alloy solidification onset of the initialinstabilityrdquo Proceedings of the National Acadamy of Sciences ofthe United States of America vol 95 no 2 pp 431ndash438 1998
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
High Energy PhysicsAdvances in
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
AstronomyAdvances in
Antennas andPropagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
International Journal of
Geophysics
Advances inOpticalTechnologies
Hindawiwwwhindawicom
Volume 2018
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
High Energy PhysicsAdvances in
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
AstronomyAdvances in
Antennas andPropagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
International Journal of
Geophysics
Advances inOpticalTechnologies
Hindawiwwwhindawicom
Volume 2018
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
![Dendrite Suppression by a Polymer Coating: A Coarse ... · Driven by theoretical predictions,[9,10] the use of solid electrolytes to mechanically inhibit dendrite growth has also](https://static.fdocuments.us/doc/165x107/5fb1fd8289d07e6b935baa40/dendrite-suppression-by-a-polymer-coating-a-coarse-driven-by-theoretical-predictions910.jpg)
