Crystal Growth Rate in Ultrathin Films of
Poly(ethylene oxide)
K.DALNOKI-VERESS,1J.A.FORREST,2M.V.MASSA,1A.PRATT,3A.WILLIAMS3
1Department of Physics and Astronomy,McMaster University,Hamilton,Ontario L8S4M1,Canada
2Department of Physics and Guelph-Waterloo Physics Institute,University of Waterloo,Waterloo, Ontario N2L3G1,Canada
3Department of Physics and Astronomy,University of Sheffield,Sheffield S37RH,United Kingdom Received8May2001;revised1August2001;accepted1August2001
ABSTRACT: Many dynamical properties of polymers,including segmental relaxation
and chain diffusion,exhibit anomalies in thin-?lm samples.We extend the studies of
thin-?lm dynamics to the case of semicrystalline polymers and present a study of the
crystal growth rate for thin?lms of poly(ethylene oxide).We used optical microscopy
and quartz crystal microbalance techniques to characterize the kinetics of crystalliza-
tion for?lms with thicknesses from40to1000nm for a range of temperatures near the
melting point.A remarkable slowing down of the crystal growth is observed at all
temperatures studied for?lms with a thickness of less than?100nm.The results can
be used to suggest reductions of the mobility of chains at the crystal/amorphous
interface.?2001John Wiley&Sons,Inc.J Polym Sci Part B:Polym Phys39:2615–2621,2001
Keywords:crystallization;thin?lms;poly(ethylene oxide);chain mobility;spheralites
INTRODUCTION
Polymer crystallinity in con?nement is of tremen-dous interest especially because of the expecta-tion that these studies can in some way elucidate fundamental issues of polymer crystallization in general.However,we must realize that because a signi?cant fraction of industrially relevant poly-mers are semicrystalline,understanding of crys-tallinity in thin?lms and at interfaces is of par-amount importance.There are various ap-proaches to con?ning semicrystalline polymers with the general consensus that the crystalliza-tion kinetics and morphology are affected.For example,the diverse phases of block copoly-mers,1–6con?nement in pores,7and the thin-?lm geometry8–19all provide evidence that the kinet-ics and morphology of semicrystalline polymers are signi?cantly affected.There are several length scales below which con?nement might be expected to affect crystallization.We would ex-pect that the kinetics of crystallization are af-fected when the con?nement results in changes in the glass-transition temperature(T g),which for polystyrene?lms on a substrate occurs at?50 nm.20Reductions in T g have also been linked to the end-to-end size of a polymer molecule21:a length scale below which the kinetics of crystalli-zation might also be affected.Likewise,when the con?ning dimensions start to approach the length scale of the lamellar thickness of the polymer crystals,d,anomalous behavior may be expected.
The multitude of block copolymer mesophases provide a tremendous opportunity for studying con-
Contribution from the March2001Meeting of the Ameri-
can Physical Society–Division of Polymer Physics,Seattle,WA
Correspondence to:K.Dalnoki-Veress(E-mail:dalnoki@
mcmaster.ca)
Journal of Polymer Science:Part B:Polymer Physics,Vol.39,2615–2621(2001)
?2001John Wiley&Sons,Inc.
2615
?nement in one-dimension(lamellae),two-dimen-sions(cylinders),and three-dimensions(spheres)at very small length scales(?10nm).The?rst of these studies were carried out by Lotz and Kovacs1 on poly(ethylene oxide)-block-polystyrene diblocks (PEO-b-PS).The same PEO-b-PS system has been studied in great detail by Cheng’s group.2,3For ex-ample,Zhu and coworkers2,3investigated crystalli-zation in a lamellar phase of PEO-b-PS where they observed changes from bulk behavior in the kinetic effects as well as carrying out detailed measure-ments of the orientation of the crystal axis within the lamellae.Loo,Register and Ryan4have applied time-resolved X-ray scattering to probe the dynam-ics of crystallization in25-nm spherical mesophases of a polyethylene block connected to a random sty-rene-ethylene-butene terpolymer block.The study revealed that the isothermal crystallization kinetics of the con?ned system was affected by comparison to a bulk system,as was the degree of undercooling required to crystallize the samples.In a similar study,Loo et al.5were able to image the microdo-main and crystallite morphologies of a poly(ethyl-ene-block-vinylcyclohexane)and study the con?gu-ration of the crystallites within these domains. Surprisingly,a study of thin?lms of a PEO-b-poly-butadiene(PEO-b-PB)diblock by Hong et al.6re-sulted in the observation that adjacent layers of the crystallites were in orientational registry while sep-arated by?10-nm layers of the PB block.The ori-entational correlation between adjacent layers must be the result of a common nucleus,and it was proposed by Hong et al.6that this is achieved through edge and screw dislocations in the mi-crophase structure.
A different approach was taken by Filippov, Doroginizkij and Vartapetyan7who con?ned PEO in cylindrical pores of active carbon with an aver-age diameter of48nm.The kinetics of crystalli-zation was studied in these samples by looking at the induction period and found to be a factor of two longer for PEO con?ned to pores as compared with the bulk.
The use of ultrathin?lms to study one-dimen-sional con?nement provides a very simple and versatile method by which the con?nement effects on crystallization can be examined.In a series of seminal articles by Frank,Despotopoulou and co-workers,8–10poly(di-n-hexyl silane)?lms as thin as?10nm were studied.Speci?cally they ob-served the following:(1)the?nal degree of crys-tallinity decreased with decreasing?lm thick-ness,h,while being almost insensitive to the sub-strates used;(2)crystallization occurs faster for thicker?lms than for thinner?lms and was com-pletely inhibited in the thinnest?lms(?15nm); and(3)these results were observed both for high and low molecular weight polymers indicating that chain dimension and entanglement cannot account for these observations.Kressler,Wang and Kammer11investigated poly(?-caprolactone)?lms in which dewetting of the polymer could be made to take place at the same time as crystalli-zation with surprising morphologies as a result of the two competing phenomena.A real-time atomic force microscopy(AFM)study by Pearce and Vancso12focused on both melting and crys-tallization of PEO.In this study,the same lateral crystal-growth rate was obtained at small length scales(AFM)and at large length scales(optical microscopy).In addition,the researchers directly observed a depletion zone of amorphous material at the melt/crystal interface.AFM was also used by Reiter and Sommer13–15to explore crystalliza-tion in PEO monolayers adsorbed onto the oxide layer on Si substrates.Perhaps the most impor-tant aspect of this work is that,in contrast to the work by Frank et al.8,they observed crystalliza-tion at monolayer length scales.In this system the area occupied by an adsorbed monolayer is greater than the area required by the crystalline phase,resulting in a limited diffusion of the ad-sorbed chains to the crystal front.This diffusion limited aggregation process results in a?ngering instability.The associated morphologies and width of the?ngers were explained in terms of a simple model and computer simulations.A very novel system was studied by Prud’homme’s group:16?lms of a miscible blend of poly(?-capro-lactone)(PCL)and poly(vinyl chloride)in which the PCL component crystallizes.They found that a surface enrichment of PCL results in dramatic changes in the growth rate of crystals depending on the distance of a nucleation site from the en-riched interface.
The main objective of this article is to examine the crystal growth rate under isothermal crystal-lization conditions in thin?lms.Other research-ers have recently performed a similar study on isotactic polystyrene(is-PS)?lms with?lm thick-nesses as low as25nm.17,18The results obtained show that the thinnest?lms have growth rates, G,that are about75%of the saturation value for thick?lms.An analysis of the data by?tting to G(h)?G(?)(1?d/h)results in a constant value for d?6nm,independent of the crystallization temperature,T c,the weight-average molecular weight M w,and the substrate used.Work from
2616DALNOKI-VERESS ET AL.
the same group found that the morphology of the crystals was also affected.19
In this article,we measure the growth rate of spherulitic crystals of PEO?lms with optical mi-croscopy for?lms as thin as40nm(the use of the term‘spherulite’is somewhat misleading because within the?lm the‘spherulite’grows laterally in two dimensions and is constrained in the?lm’s normal direction,i.e.,a crystalline disk).The studies that were performed for various?lm thicknesses,crystallization temperatures,and on two substrates reveal a reduction in G from the bulk value.Remarkably this reduction in G cor-responds to a factor of three decrease for the thin-nest?lm when compared with bulk values and is independent of the substrates used.The data analysis indicates that it is the chain-transport mechanism that is affected by the con?nement rather than the energy barrier associated with nucleating chains at the crystal front.A quartz crystal microbalance technique is also used to examine the kinetics of crystallinity and proves to be a valuable tool for further studies of crystallin-ity in very thin?lms.
EXPERIMENTAL
High molecular weight(M w?272,000),monodis-perse(M w/M n?1.14)PEO(Polymer Source Inc., Dorval Quebec,Canada)was dissolved in aceto-nitrile with polymer mass fractions ranging from 0.65to3.00%.The solutions were spin coated onto three of the following types of substrates:(1)Si with the native oxide layer present,(2)Si sub-strates onto which a110-nm layer gold was evap-orated,and(3)the gold electrode on the quartz crystals used in the QCM measurements.This study focused on?lm thicknesses ranging from40 to1000nm as measured by ellipsometry and AFM.The QCM crystal was placed in a temper-ature-controlled sample environment,and abso-lute temperatures were measured to within0.5°C.For the optical microscopy measurements a commercial hot stage(Linkam THMS-600)con-trolled the sample temperature to within the same tolerance.The bulk of the measurements were carried out in air although data were also obtained in argon as well as nitrogen to rule out effects as a result of humidity and oxygen.
Both QCM and video optical microscopy were used for the study of crystallization in the thin PEO?lms.The QCM is a very sensitive ultrasonic technique where,in this novel implementation developed by Q-sense,23the crystal with gold elec-trodes is driven at resonance(?10MHz)and then disengaged from the driving circuit.The decaying oscillation is measured with an oscilloscope,and both the resonant frequency and the decay enve-lope of the voltage signal are?t to the functional form,V(t)?exp(?t/?)cos(2?ft),where t is time,?de?nes the decay envelope,and f is the frequency of oscillation.The energy dissipation per cycle,D (also known as the inverse of the Q-factor),of the crystal/sample system is given by D?(2?f?)?1 and depends on the shear loss in the system.For the study of crystallization,the dissipation factor provides valuable information.The shear loss modulus of the crystalline and liquid phases are very different,and hence the change in the dissi-pation is simply proportional to the area of the ?lm on the active area of the quartz crystal that remains liquid:?D?A(liq).The QCM used in this manner is very sensitive to the crystallinity of the sample,with the ability to track the crystal volume fraction in?lms as thin as tens of nano-meters.This is only possible because of the tre-mendous sensitivity provided by the Q-sense in-strument.In this study,we focus on the isother-mal crystal-growth rate and unfortunately although the QCM is a very sensitive tool,the crystal-growth rate cannot be determined inde-pendently.This is because the QCM measure-ment of the dissipation depends not only on the crystal-growth rate but also on the number of crystals on the active area.
Video optical microscopy provides a very straightforward approach to probing the growth rate of crystals provided the boundary between the crystal and the liquid can be visualized and followed as a function of time.Dark?eld micros-copy provided the most contrast.In this mode,the sample is illuminated off-normal to the sample surface,and most of the light seen through the microscope is scattered by the crystals.This gives the appearance of white crystals in a black back-ground as seen in Figure1.The only drawback of this optical method is that it was found to be limited to?lms with a thickness,h,greater than ?40nm as a result of the lack of contrast. RESULTS AND DISCUSSIONS
Although the QCM technique was not used for the bulk of the data on which this article is based,the technique did prove to be very sensitive and pro-vided complimentary information.In Figure2we
CRYSTAL-GROWTH RATE2617
show the dissipation,D ,for a 200-nm ?lm as a function of time for ?ve different isothermal crys-tallization experiments.For these measurements the samples were annealed for 20min at 80°C,a temperature above the ideal melting tempera-ture,T m ?68°C,followed by a rapid quench to the crystallization temperature,T c .We clearly see the expected behavior:initially there is no change in the dissipation during the induction period until nuclei are formed at which point D decreases as a result of the growth of the spheru-lites;after some time the spherulites impinge on each other,and the rate of change of crystallinity slows down.For higher degrees of undercooling the process is observed to occur much faster be-cause the driving force to crystallize is larger (when the crystallization temperature is changed from 54to 50°C almost an order of magnitude change in the timescale is seen).
In the second type of experiment,the sample was cooled at some speci?ed ramp rate until the sample had fully crystallized and then heated again at the same rate.Figure 3depicts the re-
sults of such an experiment for four different ramp rates ranging from 10to 1°C/min.For the faster ramp rates,the liquid can be supercooled further before the sample crystallizes.Upon heat-ing the temperature at which the crystals melt is found to be the same.The ease of using the QCM to carry out such measurements for even very thin ?lms makes this an ideal tool for the study of crystallinity in con?ned systems.
This article focuses on the crystal-growth rate rather than the more complicated issue of crys-tallinity in con?nement which is dependent on both the crystal-growth rate and the nucleation rate of new crystals and may also be affected by con?nement.7Optical microscopy studies,as shown in Figure 1,can be used to obtain the radius of the spherulite as a function of time and hence the growth rate,G .In Figure 4we plot the change in the radius as a function of time for a 200-nm ?lm.For these isothermal crystallization experiments,the samples were annealed at 80°C for 20min before quenching to the crystallization temperature T c ?T m .The results for six different values of T c are illustrated in Figure 4,and
in
Figure 1.Optical micrographs of a 200-nm PEO ?lm.The time interval between images is 20s,and each image is 620?m wide.The sample is crystallized isothermally at 48
°C.
Figure 2.QCM measurement of the dissipation fac-tor as a function of time for a ?lm thickness of 200nm and ?ve isothermal crystallization
temperatures.Figure 3.The QCM dissipation factor as a function of temperature for four different temperature ramp rates for a 200-nm ?lm.
2618DALNOKI-VERESS ET AL.
each case we obtain a best ?t value of the slope G .As with the QCM results,we see the expected behavior,that is,an increase of G with an in-crease in the degree of supercooling.In this way the growth rate was obtained for nine ?lm thick-nesses (40nm ?h ?1000nm)and at least seven crystallization temperatures for each ?lm thick-ness ranging from 42to 56°C.The results for six ?lm thicknesses are represented in Figure 5where we plot the growth rate as a function of the T c .A surprising result is immediately obvious:decreases in the ?lm thickness result in reduc-tions in the growth rate.Similar results have been observed previously in studies of isotactic polystyrene.17,18The decrease in G is signi?cant and corresponds to approximately a factor of
three difference between the two thickness ex-tremes (as compare with isotactic polystyrene where the growth rate was found to be ?0.75of the bulk value for similar thickness values 17).A technical dif?culty observed with these mea-surements was systematic changes in the sample upon thermal cycling.Initially many crystalliza-tion runs were performed on the same sample by reannealing at 80°C for 10min.It was found that under identical conditions the growth rate de-creased in subsequent cycles (see Fig.6).24Cur-rently we do not fully understand this result,although preliminary studies indicate that this is due to surface roughness of the melt that is not fully annealed away before recrystallizing the sample.This claim is consistent with the obser-vation that longer annealing times between cycles diminished the reduction in the growth rate.To avoid this problem we used only ?rst-cycle data as well as longer annealing times (20min).With these precautions,reproducible data could be ob-tained that were independent of the atmosphere in which the experiments were carried out (air,nitrogen,and argon).All the data used in the following analysis and shown in the ?gures are obtained in this way.
We now return to the results of Figure 5.To fully characterize these results,the expression for the temperature dependence of the crystal-growth rate can be used
G ?exp ???F *?kT ?exp ???G *?kT ?;
where ?F *is the activation barrier to chain trans-port,and ?G *is the free-energy barrier for
nucle-
Figure 4.The change in the radius as a function of time for six different isothermal crystallization temper-atures,h ?200
nm.
Figure 5.The growth rates obtained for six ?lm thicknesses at various isothermal crystallization tem-
peratures.
Figure 6.Growth rate in subsequent cycles on the same sample.Isothermal crystallization at 46°C,h ?50nm.
CRYSTAL-GROWTH RATE 2619
ation.25,26For a small temperature range close to T m we can ignore the temperature dependence of the activation barrier to chain transport (i.e.,over a small temperature range at T ??T g ,chain mobility is constant with T ),and using 25?G *?1/(T m ?T )we rewrite the crystal-growth rate as
G ?G ?exp ??b ?T ?T m ?T ??.
This expression has the important characteristic that the crystal-growth parameter G ?is indepen-dent of the temperature (for the small tempera-ture range studied)and describes chain trans-port,whereas b is a parameter that characterizes the energy barrier to the nucleation of a new polymer segment at the crystal surface.Remark-ably,the data for all ?lm thicknesses can be ?t very well using this expression with a single value of b ,whereas the crystal-growth parameter G ?(h )is ?lm-thickness dependent (see solid lines in Fig.5).In the data analysis we used T m ?68°C and obtained a best ?t value for b ?(4.0?0.3)?104K 2.The best ?t value of b was ?xed for the ?ts to obtain the values of G ?(h )plotted in Figure 7.The ?gure illustrates unambiguously the factor of three decrease in the crystal-growth rate for the thinnest ?lm for the entire temperature range studied.The excellent ?t to the data obtained with a single value of b indicates that the energy barrier to the nucleation of a new polymer seg-ment at the crystal surface is unaffected by the
con?nement for the ?lm thicknesses studied (i.e.,the drive to crystallizing is not dependent on h for the range of h studied).In contrast to b ,the growth parameter G ?(h )is strongly ?lm-thickness dependent and is indicative of anomalous chain transport at the crystal/amorphous interface in the thin PEO ?lms.A somewhat different analy-sis of the data obtained for isotactic polystyrene,it-PS,resulted in similar claims by Sawamura et al.17To verify if the observed behavior was the result of some substrate interaction,the measure-ments were repeated on a gold substrate with identical results as shown in Figure 7.In addi-tion,measurements of the crystal-growth rate on Si substrates annealed and crystallized in argon are also shown in Figure 7.Within the experi-mental uncertainty,the data obtained for all mea-surements are consistent regardless of the sub-strates used and the atmosphere (air,nitrogen,and argon)in which the experiments were carried out.
The simple analysis of the data implies that it is the chain transport that is affected.Unfortu-nately it is not possible for us to provide a mech-anism responsible for the anomalous chain trans-port.For studies of the T g of polystyrene,both on substrates and in a free-standing geometry reduc-tion of T g with decreasing ?lm thickness was ob-served.This seems to oppose the inhibited chain transport for thin ?lms observed in this PEO study as well as similar results obtained by Sawamura et al.17for it-PS.However,these ob-servations should not be taken as contradictory because the segmental mobility probed in a T g experiment and the chain mobility associated with a chain attaching itself to a crystal surface are vastly different.
SUMMARY
In this article,we report on the kinetics of crys-tallinity in thin ?lms of PEO.A QCM technique was used to track the crystallinity in ?lms that are only tens of nanometers https://www.doczj.com/doc/9117154348.html,ing the QCM it was possible to track the kinetics of crystallin-ity in isothermal experiments as well as the de-gree of supercooling with various temperature ramp rates.Optical microscopy was used to ob-serve reductions in the growth rate of spherulites when con?ned to a thin-?lm geometry.As the ?lm thickness was reduced from 1000to 200nm,no change in the crystal-growth rate,G ,could be measured;however,further decreases in the
?lm
Figure 7.Temperature-independent growth rate (see text for de?nition)as a function of the ?lm thickness.The solid circles are the data for PEO ?lms on a Si substrate with the native oxide layer present,whereas the open squares are for measurements carried out entirely in an Ar atmosphere.The open circles are for measurements of the growth rate of PEO on a Au substrate.
2620DALNOKI-VERESS ET AL.
thickness to40nm caused a factor of three de-crease in G.This result was observed over tem-peratures ranging from42to56°C and found to be independent of the substrate interaction for the two substrates studied as well as independent of the atmosphere in which the experiments were carried out.Although a cause for the reductions in G could not be provided on the basis of the current data,a simple analysis indicates that it is the chain transport that is affected by the con?ne-ment and not the energy barrier associated with nucleating additional chain segments at the crys-tal front.
The authors thank Profs.S.Z.D.Cheng,R.A.L.Jones, Y.Miyamoto,and R.E.Prud’homme for the helpful discussions.This work was funded in part by EPSRC of the U.K.and NSERC of Canada. REFERENCES AND NOTES
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CRYSTAL-GROWTH RATE2621
基于Crystal Ball 软件对测量不确定度的评定 1240410114 王颖测量结果与被测量真值的一致程度被定义为准确性。但是实际上不存在完全准确无误的测量，因此通常在给出量值结果的同时通常给出适应于实际需要的不确定度。如果没有对不确定度的表述，所进行的测量的被测量对象的质量就无从判断，从而导致测量的结果值不具备充分的实用价值。测量的结果值的准确，是在一定的不确定度、误差允许误差范围内的准确。 一）基本概念 测量不确定度的概念最早是有国外引入，一般译为：与测量结果相联系的参数，用来表示赋予被测量对象值的分散性的特征。它最早跟我们熟悉的误差的概念相似。测量不确定度的前提是当我们在重复性条件下，对具有稳定特征的被测量对象X独立的进行了n次重复测量实验，在这一系列测量实验过程中，通过n个结果按公式计算出的，第i次结果xi的实验标准差E（xi)，xi虽然是指第i 次测量的结果，但是它的实际含义是：任一次的测量结果。表明不确定度s(xi)=u(xi)是这个测量序列中任意一次测量结果的不确定度。如果在相同的相同的、重复条件下再进行测量，得到的结果xi 的标准不确定度仍然是E(xi。 二）测量不确定度评定的步骤
1.识别不确定度来源。对测试结果测量不确定度来源的识别应该首先从分析测量过程开始，并且要对测量方法、测量系统和测量程序作详细研究和熟悉，如果可能要画出测量系统原理图和测量流程图。不确定度来源一般有：对被测量的定义不完善；实现被测量的定义的方法不理想；选取测量样品的典型性不够；对测量过程中受外部环境影响的因素识别不完整等因素引起。 2.建立模型。当被测量对象Y(即我们期望的输出量)由N个其他因素X1，X2，…，XN(即输入量)，通过函数关系f来确定时，则Y = f (X1 , X 2 ,L, X N )称为测量模型或数学模型。式中大写字母表示测量的符号f 为测量函数。如果输入量Xi 的估计值为Xi，被测量对象Y 的估计值为y，则测量模型可建立为：y = f (x1 , x 2 ,L, xN ) 3.标准不确定度A类和B类分量的计算。A类不确定度分量的评估（对观测序列所进行统计分析作出的评估）。a)对输入量Xi进行独立的n 次测量，测量结果为：x1、x2……xn， 单次测量结果的标准差为：估计值的标准不确定 度为：，由于B 类的使用条件与A类不同，因此B 类不确定度分量的评估与A类也不同，B类评估时，输入量的估计量Xi不是由重复观测得到时，其标准偏差可用对Xi的信息来进行评估。B类评估的信息来源可来自：仪器设备的校准证书的说明、生产厂商标示的说明书、使用的检测依据的标准、引用手册的参考数据、以前
Crystal Ball实验操作过程 实验一： 一、数据录入与导入 双击CB快捷方式图标或直接打开Excel打开软件。前面提到过Crystal Ball软件是在Excel里的一个插件，所以双击打开后是Excel的界面，如下图： 图 1 用户可以在该界面中直接录入数据，也可以左击右上角的符号，选择打开，将原有Excel表格中的数据直接导入到带有Crystal Ball插件的电子表格中。 二、拟合分布 图2 (1)对数据进行标准化处理(减少原数据相互间的距离对拟合分布的影响) 通过Average计算每个分布工程样本数据的均值,然后各个样本数据除以相
应的均值,对数据进行标准化处理。 （2）拟合分布 选取表格区域，点击工具栏上“Run-Tools-Batch Fit”,如图3所示。 图3 在操作对话框中，选择“next”,至图4对话框对相应命令进行选择,可得到拟合过程的相关数据。 图4 注：对于卡方检验，水晶球软件计算p值，p值大于0.5一般表示紧密拟合； 对于科尔莫格洛夫-斯米尔诺夫检验，一般地，小于0.03的K-S值表明良好拟合； 对于安德森-达林检验，小于1.5的计算值一般表明拟合优良。 实验二:
一.按照实验一的操作,先将数据在Crystal Ball软件打开. 二、假设单元格概率分布的定义及相关操作 输入数据后，进行随机变量假设单元格概率分布的定义。这里假设使用悲观时间的单元格来进行概率分布的定义。（注：对于假设单元格的选择，并无太多的限制，因为定义各种概率的分布，是由相应的参数确定的，因此选择的假设单元格不同对结果并没有影响。）有一点需要注意的是，选择假设单元格时，该单元格应当是一确定的数字，而不能是公式. 选定单元格（如单元格I2）后，点击工具栏上的，随即弹出图5，CB 软件中提供22种不同的分布可供选择，根据实验任务书的要求，第一和第二项分部分项工程服从三参数beta分布，因此，选择BtaPERT分布，并填入相应参数，即可完成对“基坑支护挖土方”的定义，如图6所示。同理可完成其它分布的定义。 图5 图6 由于第3~8项同为三角分布，因此当完成第3项的定以后，选定I4单元格
Crystal Ball 介绍 Crystal Ball(CB)是基于PC Windows平台而开发的简单且非常实用的风险分析和评估软件。面向各类商务、科学和技术工程领域，用户界面友好,是基于图表进行预测和风险分析。CB 在微软Excel 应用软件上运行，使用蒙特卡罗（Monte Carlo）模拟法对某个特定状况预测所有可能的结果，运用图表对分析进行总结，并显示每一个结果的概率。除了描述统计量、趋势图和相关变量分配，CB还进行敏感性分析，让用户决定真正导致结果的因素。如今 CB 已是全世界商业风险分析和决策评估软件中的佼佼者。Crystal Ball专业版是市面上以Excel为本的风险分析及预测工具中最全面的套装软件。其功能和特点不仅早已得到广大用户的认同，并获得许多正在考虑购买相关软件产品新用户的青睐和首选。85%<<财富>>评出的全球500强大企业中早已有400家使用 Crystal Ball 软件作为他们进行商务决策，项目投资风险分析的工具。再者,美国前50名最佳MBA 商学院，已有40所也用Crystal Ball作为教研和商业性课题的工具。用户之一世界着名的哈佛大学商学院把 Crystal Ball 列为可用于计划金融的软件 (Project Finance Software)。因为财政计划,金融投资方面的风险分析是CB 软件功能的一部分。 Crystal Ball之前是美国Decisioneering公司的产品，Decisioneering在2007年被Hyperion公司收购，Hyperion 公司之后又被Oracle收购，所以Crystal Ball目前的发行商是Oracle。 Crystal Ball的用途： DFSS，过程研究，过程优化，现有过程的模拟改变，公差分析，设计分析，原料筛选，容量设计，资源分配与存货优化，约束后的项目筛选，预防性维护优化，成本预算，可靠性分析，排队过程分析，建筑项目资金预算的偶然性分析，商业过程模拟，工程设计与预测，供求预测，制造供应链问题的减少与存货控制，新产品商品化的资金模型。 模拟的意义 当我们使用模拟这个字时，代表我们利用分析模型来仿真现实生活的系统。过去仿真软件过于偏重复杂数学造成操作困难。Crystal Ball工作表风险分析结合工作表呈现方式与自动分析模拟，可以清楚的展现因为变量变异造成模型产出的各种情况。如果没有增加仿真功能，那工作表充其量只是揭示单一结果与最一般化的情境。工作表模拟最常用的方法就是蒙特卡洛法，它可随机产生变量在不同情况下的模型结果。 蒙特卡洛模拟 蒙特卡洛模拟一开始主要运作于分析赌博游戏。诸如轮盘、骰子、拉吧等。蒙特卡洛可以模拟这些赌博中的随机行为。当你掷骰子时，你知道共有一至六的数字可能会出现，但是你不知道一个规则。他就像企业主面对问题时，可能知道问题引发的结果与过程，却无法了解每一个变量的严重程度。（例如：利率、员工、股价、存货及来电率） 自行于工作表内选定变量的分配类型 针对每个不确定变数，你可以自行设定相关机率分配。您可以依据该变量所处的环境选择分配的类型，分配类型如下： 您可以将工具栏加入EXCEL工作表中，当然您必须知道这些等式所代表的分配。透过Crystal Ball, 软件可自动为用户计算方程式。Crystal Ball甚至可以藉由过去的历史数据来修正分配。
Monte-Carlo Simulation with Crystal Ball? To run a simulation using Crystal Ball?: 1. Setup Spreadsheet Build a spreadsheet that will calculate the performance measure ., profit) in terms of the inputs (random or not). For random inputs, just enter any number. 2. Define Assumptions—., random variables Define which cells are random, and what distribution they should follow. 3. Define Forecast—., output or performance measure Define which cell(s) you are interested in forecasting (typically the performance measure, ., profit). 4. Choose Number of Trials Select the number of trials. If you would later like to generate
the Sensitivity Analysis chart, choose “Sensitivity Analysis” under Options in Run Preferences. 5. Run Simulation Run the simulation. If you would like to change parameters and re-run the simulation, you should “reset” the simulation (click on the “Reset Simulation” button on the toolbar or in the Run menu) first. 6. View Results The forecast window showing the results of the simulation appears automatically after (or during) the simulation. Many different results are available (frequency chart, cumulative chart, statistics, percentiles, sensitivity analysis, and trend chart). The results can be copied into the worksheet. Crystal Ball Toolbar: Define Define Run Start Reset Forecast Trend
软件介绍 CrystalMaker：晶体和分子结构可视化软件 探索晶体世界 从金属到沸石，苯到蛋白质, CrystalMaker是了解晶体和分子结构的最简单方法。 什么是CrystalMaker? CrystalMaker软件是一款在创建、显示和操作各种晶体分子结构中屡获好评的软件。CrystalMaker在生产力方面提供了一个流线型的工作流程，您只需把您的数据文件拖拉到程序中便可即时显示照片般逼真的色彩。用鼠标就可以实时操作晶体结构。多视角"bookmarks"和撤销次数鼓励您探索和发现——理想的教学和科研软件。 快速创建晶体和分子结构！ 使用CrystalMaker软件，您可以轻松快速创建任何晶体或分子结构。内置的对称处理和美观的空间群浏览器可得到晶体的日志，并且该程序会自动生成所有的键和多面体。CrystalMaker 提供了广泛的模式类型，包括传统的“ball-and-stick”, space-filling, polyhedral, wireframe 以及thermal ellipsoids模式。使用photo-realistic simpler和line-art display选项，每种模式类型都可以被广泛地定制。 综合数据的输入和输出 负载来自于超过15+ 格式的结构数据，包括Cambridge Structures Database, Protein Data Bank, CIF, GSAS, SHELX等。您可以操作几乎无限的原子数量。使用独特的"Depth Profiling"工具，快速扫描大规模结构中有用的，从而能从计算机模式中得到理想的特征结果。 大量的输出选项使您能跟其他程序共享数据、保存结构数据、键长、协调环境，甚至用您的数据创建网页。 出色的3D图片 深度渐变和视角转换，加上优美的三维立体结构，使您能看到照片质量的图片。高分辨率打印并把图片保存成各种文件格式，同时您还可以定义图片的大小。使用深度剖析和测量处理大规模的结构。完整的误差传递、集群壳和协调网络可视以及强大的输出选项，自动生成键长和多面体。 实时操纵与测量 用鼠标拖拉、键盘或工具栏，便可查看网格矢量或平面矢量。连续绘图范围设置可以设置数百万个原子，健和多面体。 移动、分离、复制、隐藏和删除原子组。隐藏或修复分子片段——分离单个分子。可在任何方位显示平面格。切割晶体结构以便研究其表面或内部平面。或把一个晶体或分子结构置于另一个结构中。 预览协调环境，集群和表面。列出键长和屏幕上的柱状图或保存到磁盘里。在屏幕上测量键长，角度，以及扭转角。在平面和向量间计算角度。
Crystal Ball 模拟基础教程 利用Crystal Ball 进行计算机仿真 学习目标13.2 个案研究：佛莱迪报童问题（13.1节） 13.3–13.19 竞标建设计划（13.2节） 13.20–13.24 项目管理：信用建设公司（13.3节）13.25–13.32 现金流量管理：沼泽地黄金岁月公司（13.4节） 13.33–13.37 财务风险分析：久大发展公司（13.5节）13.38–13.42 运输业收入管理（13.6节）13.43–13.48 选择合适的分配（13.7节）13.49–13.68 利用决策表做决策（13.8节） 13.69–13.84 学习目标 在读完本章后，你应该能够： 1. 描述Crystal Ball在计算机仿真中的角色。 2. 利用Crystal Ball来解决Excel软件包所无法执行的各类基本计算机仿真。 3. 解释利用Crystal Ball于计算机仿真中的结果。 4. 在获得预期的准确度水平后，利用Crystal Ball的特色来停止计算机仿真。 5. 描述当使用Crystal Ball时可以搭配计算机仿真的机率分配之特色。 6. 利用Crystal Ball程序辨识出符合历史数据的连续分配。 7. 利用Crystal Ball的特色来产生一些帮助决策的决策表和趋势图。 报童佛莱迪 佛莱迪在某大城市里主要市区经营一家报摊。 佛莱迪贩卖各类的报纸和杂志，其中最贵的报纸为财经日报。 财经日报相关的成本资料： –每份报纸的成本为1.50美元 –每份报纸的售价为2.50美元 –没售出的报纸，每份报纸可以获得0.50美元的偿还金 财经日报的销售资料： –佛莱迪每天的销售量介于40到70份之间。 –销售数量介于40到70份之间任何数值的频率相同。 运用仿真之电子表格模式 Crystal Ball的应用 利用Crystal Ball来进行计算机仿真有四个步骤： –定义随机输入栏。 –定义输出栏来预测。 –设定执行偏好。 –执行模拟。 步骤1：定义随机输入栏 随机输入栏是拥有随机数值的输入字段。 需要输入储存格的是假设的机率分配而非一永久的数值。 Crystal Ball将每个随机输入栏称作假设栏（assumption cell）。 定义假设栏的步骤 ?点选选定的字段。 ?假如字段没有数值，输入任何一个数字。 ?点选「Crystal Ball」标签（Excel 2007）或工具列（Excel较早版本）的「Define Assumption」
风险管理软件CrystalBall操作指南(英文版)(doc 16页)
Monte-Carlo Simulation with Crystal Ball? To run a simulation using Crystal Ball?: 1. Setup Spreadsheet Build a spreadsheet that will calculate the performance measure (e.g., profit) in terms of the inputs (random or not). For random inputs, just enter any number. 2. Define Assumptions—i.e., random variables Define which cells are random, and what distribution they should follow. 3. Define Forecast—i.e., output or performance measure Define which cell(s) you are interested in forecasting (typically the performance measure, e.g., profit). 4. Choose Number of Trials Select the number of trials. If you would later like to generate the Sensitivity Analysis chart, choose “Sensitivity Analysis” under Options in Run Preferences. 5. Run Simulation Run the simulation. If you would like to change parameters and re-run the simulation, you should “reset” the simulation (click on the “Reset Simulation” button on the toolbar or in the Run menu) first.