SAMPLE ANSWERS TO HW SET 3A

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1 SAMPLE ANSWERS TO HW SET 3A Cpied belw are sme examples f (mre r less) successful answers t the HW prblems that were riginally due n Sept. 6 th. I selected these examples t give everyne an idea n hw t slve each prblem- dn t read anything int why I chse what examples I did. Mstly, I ve selected answers that seem straight-frward. 1. Calculate the rate f rise f spherical bubbles f different sizes (frm 10 micrns t 1 cm) in a mlten glass tank, glass density.5 g/cm 3 and a viscsity f 10 P. Hw lng will it take bubbles f different sizes t rise 100 cm under these cnditins? Example 1: Use f the frmula V = d * g * (ρ bubble ρ liquid )/(1*η) gives the abve curve fr the glass melt f density.5 g/cm 3 and viscsity f 10 P, assuming the bubbles have the density f dry air (apprximately g/cm 3 ). The velcity diameter curve shws a quadratic relatinship, since the velcity varies with the square f the bubble diameter. Sme rise velcities are given in the table belw: Diameter (cm) Rise Velcity (cm/s) E Since velcity is equal t the rise distance divided by the rise time, rearrangement f the velcity equatin gives the frmula t rise = (1 * y * η)/(d * g *(ρ bubble ρ liquid )), where y is the

2 vertical rise distance f 100cm. The rise time thus varies inversely with the square f the bubble diameter, leading t the curve shwn belw. Example : The velcity f bubble rise was calculated using the crrected Stke s Frmula belw where: v=rate f rise r=bubble radius ρ = density f the glass ρ =density f the air inside the bubble g = acceleratin f gravity η = viscsity f the glass η = viscsity f the air inside the bubble v = r g ( ρ ρ ) 3η + 9η η + 3η 3η The time fr each bubble t rise 100cm was then calculated using the velcity.

3 ρ (g/cm 3 ) η(p) ρ (g/cm 3 ) η (P) d (cm) E E r (cm) v (cm/s) t (s) E-06 1.E E E E E E E E E E E E-04.50E E E E E E-04 1.E E E E E E E E E E E E-0.50E E E E E E-0 1.E E E E E E E E E E E E+00.50E E E E E E+00 1.E+01 In general, almst everyne nailed this prblem. The mst egregius mistakes invlved using a significant density fr the gas bubble- it reality, yu culd ignre the gas density and get a reasnable answer. A few had sme prblems with units- and s had answers ff by sme factr f ten. One r tw prvided answers that I culdn t really decipher. Average scre n the 6 answers submitted: 3.1/5

4 . Cnsider a sda lime silicate glass fiber in a Littletn sftening pint experiment. The surface tensin f the glass in a dry atmsphere is 300 mn/m and the measured Littletn sftening pint is 750 ºC. When the fiber is equilibrated in a wet atmsphere, the surface tensin is reduced t 00 mn/m. By hw much des the apparent Littletn sftening temperature change? Example 1:

5 Example : The equatin t describe the viscsity f a fiber in a fiber elngatin experiment is: η = Lp + p γp ρg r dp 3 dt where η is the viscsity, L is the length f fiber utside f the furnace, p is the length f fiber inside the furnace, ρ is the glass density, g is the acceleratin f gravity, γ is the surface tensin, r is the average radius f the fiber, and dp/dt is the fiber elngatin rate. The Littletn sftening pint experiment has set values fr many f these variables, such that η = Pa s when L = m, p = m, r = mm, and dp/dt = 1 mm/min. If a glass with a surface tensin f γ = 300 mn/m is subjected t these cnditins (and if we assume the average acceptable radius f 0.35 mm), we can back calculate the density f the glass frm the abve equatin t be.738 g/cm 3. ρ = dp γp 3η + dt r + g Lp p = 665 ( Pa s) ( 981. m s ) 1 ( 03. )( 010. ) 0 001m Nm m + 60sec m m m+ ( m) = kg m 3 If this same glass was then equilibrated in a wet atmsphere such that the surface tensin was lwered t 00 mn/m, the apparent viscsity wuld decrease t Pa s. Lp + p η = 3 γp ρg r dp dt ( ) m m m = = 50801Pa s = 10 Pa s 3 ( 738kg m )( 9. 81m s ) m sec 1 ( 0. 00N m )( m) m Examining the viscsity temperature relatinship f a typical sda lime silicate glass will give us the apparent Littletn sftening temperature in the wet cnditins. We can assume that the viscsity fllws a VFT relatinship, described by the fllwing equatin:

6 ΔH η = η exp RT ( T) and we can slve fr the pre expnential factr (η ), activatin energy f viscsity (ΔH), and cnstant (T ) by slving simultaneus equatins with knwn values f viscsity and temperature. Fr a typical sda lime silicate cntainer glass, Varsh gives the fllwing viscsitytemperature values: η (Pa s) T (ºC) Plugging these values int the VFT equatin and slving simultaneusly yields the fllwing values fr the unknwn variables: Variable Value η (Pa s) 1.913E-05 ΔH (J/ml) T (ºC) 41.6 We can plug these values back int the VFT equatin, alng with a viscsity value f η = Pa s, t btain the temperature at which the given viscsity will be bserved. The result is ºC., T = T + ΔH = 416. C + R ln η η J / ml ( J / mlk) ln 10 ( E 5) = C Therefre, the apparent Littletn sftening temperature will decrease by ~ 3.4 ºC when the surface tensin f the SLS glass drps frm 300 mn/m t 00 mn/m when equilibrated in a wet atmsphere. This prblem was tricky. The best apprach, I think, was t cnsider hw the apparent viscsity wuld change with decreasing surface tensin (it wuld appear t decrease because a lwer surface tensin ffers less resistance t the fiber elngatin) and then t estimate the temperature change that wuld prduce an equivalent viscsity change. These tw examples tk this apprach. The answers are a bit different, principally because f assumptins abut η(t), but the apprach is slid. Average scre n the 6 answers submitted: 0.8/5

7 3. Review the 1995 Science article by Angell, then describe and explain in detail the relatinship between the change in heat capacity at T g and the fragility characteristics f the crrespnding melt viscsities. On the Angell fragility plt why d all nrmalized viscsity curves cnverge at an apparent viscsity f ~10 4 P as T g /T 0? Example 1

8 The first part f the answer discusses the effects f structural changes in supercled melts near T g n changes in heat capacity and viscsity. Mst answers I received were similar, but mst peple had prblems with the secnd part f the questin. I like the last part f the answer shwn abve. Turns ut that the viscsity (at rm temperature and pressure f cmmn gases als fall within the range Pa s range, the same range fr which the VFT extraplatins fr silicate melts cnverge when T. (See Russell et al., Amer. Mineral (003). Example The Angell plt f viscsity versus nrmalized temperature shwing strng and fragile liquid behavir is shwn belw (frm C.A. Angell, Frmatin f glasses frm liquids and biplymers, Science 67 [506] (1995)). Glass transitin regins are ften described by the temperature f nset C p increase, ΔC p, during heating. The heat capacity f a material is a measure f a heat energy required t increase the temperature f a quantity f material by a certain temperature interval Q Cp = δ, and is affected nt nly by the mlar mass f the substance, but als by the dt p available degrees f freedm and the bnding within the substance. Therefre, structural changes will greatly affect heat capacity. Strng melts are characterized by having a strng resistance t structural change. As the temperature is increased ver a wide temperature range abve the T g, little rerganizatin f the structure ccurs. Often, these glasses have structures similar t that f the crrespnding crystal. The lack f structural rearrangement and strng crrespndence t the crystalline frm

9 result in Arrhenius behavir f the viscsity with respect t temperature and small changes in heat capacity as the melt passes thrugh the T g range. The structures f fragile liquids, hwever, are very sensitive t temperature changes and underg mre substantial rearrangements upn heating by way f changes in crdinatin and rientatin f the structural units. Fragile liquids ften have structures that differ greatly frm the crrespnding crystalline frm. This cnsiderable structural rerganizatin upn heating thrugh the glass transitin regin results in nn Arrhenius behavir f the viscsity f the melt and a large change in the heat capacity. In summary, strng liquids experience little structural rerganizatin upn heating thrugh the T g regin and thus exhibit smaller, mre gradual changes in viscsity ver a wide temperature range abve T g. The viscsity temperature relatinship demnstrates an Arrhenius dependence and the glass transitin regin is assciated with nly a small change in heat capacity. Fragile liquids underg larger structural rearrangements upn heating thrugh the T g regin and thus exhibit mre abrupt changes in viscsity with respect t changes in temperature. The viscsity temperature relatinship is nn Arrhenius and the glass transitin regin is assciated with a substantial change in heat capacity. On Angell s fragility plt, all nrmalized viscsities cnverge as T g /T 0 (T ). As the temperature increases t infinity, the liquids will all vaprize and will exist in gaseus frm. This cnvergence implies that all gases, regardless f the atmic r mlecular make up, will have the same viscsity (~ 10 4 P) at infinitely high temperatures. This phenmena is als demnstrated by the VFT equatin fr viscsity, η = η exp B T T. Accrding t the abve equatin, the viscsities f all substances will apprach η η as T, regardless f the cnstants B and T, which are dependent n the material cmpsitin. This implies that η is the same fr all materials and is equal t 10 4 P (r 10 5 Pa s). The answer t the T g /T 0 part f this last answer is similar t what I received frm mst peple. Hwever, if yu did sme digging, yu culd find a cuple f explanatins fr why the limiting viscsity is n the rder f 10-4 P (r Pa s). Fr example, in the Eyring paper I referenced in my viscsity lecture (J. Chem Phys., 4 83 (1936)),, the fllwing equatin is derived fr viscsity (η): η=(nh/v)exp(e 0 /kt) (19) where N is Avgadr s number, h is Planck s cnstant, V is mlar vlume, and E 0 is the activatin energy fr viscus flw. Eyring uses a ballpark value fr mlar vlume f 40 cc/mle and s ends up with the apprximate expressin fr viscsity:

10 Frm (0), η 10-4 Pa s as T. η 10-4 exp(e 0 /kt) (Pa s) (0) A secnd way t cnsider the limiting viscsity is t use Maxwell s relatinship (η=g τ, where G is the bulk shear mdulus, ~10 10 Pa, and τ is the relaxatin time, nw defined as the perid fr lattice vibratins, abut sec). If we assume that relaxatin rates are limited by vibratinal rates, then the minimum viscsity will be 10-4 Pa s. See CA Angell, J. Nn-Cryst. Slids, , (1991); CA Angell et al., J. Appl. Phys., (000) fr sme mre details. Average scre n the 6 answers submitted: 0.0/5 4. Frm yur review f the literature, identify a family f glasses (r representative cmpsitins) that pssess glass thermal expansin cefficients in the range x 10 7 /ºC and glass transitin temperatures belw 450ºC. Such glasses culd be used fr seals t a variety f metals. What are the advantages r disadvantages fr using yur cmpsitins fr such applicatins? Example 1 Glasses in the systems 0.MO 3 0.3P O 5 (0.5 x)pbo xpbcl (x > 0.) [1] and xpbf (1 x)[pbo:teo ] [] psses the prperties specified abve. The intrductin f halide ins int the glass is accmpanied by imprtant changes in the glass prperties. It can increase the cnductivity by increasing bth carrier cncentratin and mbility which is an advantage in the use f these glasses as metal seals. In these systems, the strng cvalent P O and M O and Te O bnds in the glassy netwrk are prbably replaced by apparently weaker inic bnds f the Pb + O type. Increasing PbCl and PbF cntent strngly affects the transfrmatin temperature T g f these glasses. T g decreases gradually with the replacement f PbO by PbCl r PbF, which is shuld be crrelated with breaking f the netwrk, cntributing t its lsening and cnsequently decreasing the bnd strength. This bservatin is cnsistent with the decrease in hardness number. The cntinuus decrease f the hardness number fr these glasses reveals that the fragility nature f the netwrk may be increased (i.e. a weaker structure btained). This behavir can be disadvantageus fr the applicability f these glasses as metal seals since the hardness f the

11 glass is ften equated with its resistance t abrasin and may determine the durability f the glass during use. An pen, less rigid r lse structure als favrs an increase in the thermal expansin. Therefre, the increase f thermal expansin with increasing PbCl r PbF cntent can be ascribed t the elngatin r rupture f sme bnds such as P O P, M O M and/r P O M in additin t Pb O P linkages due t accmmdatin f halide ins. The increasing thermal expansin may als limit the applicability f these glasses as metal seals in certain temperature ranges. [1] Hassan A. K., J. Phys.: Cndens. Matter 11, 7995 (1999). [] El Damrawi G., Phys. Status Slidi A 177, 385 (000). I included this questin as a bit f a thrw-away- there are hundreds f examples in the literature f glasses with the requisite prperties- including a few papers f my wn. (*N* bnus pints were given t anyne wh cited them!) I was surprised that a number f yu did nt find an apprpriate reference, r if yu did find ne, yu didn t prvide a useable citatin. I was als trying t find ut what yu knw abut glass prperties in general. Average scre n the 6 answers submitted: 0./5 Overall Hmewrk Scres: Average: 84. (standard deviatin = 11.5) Range: I have submitted my scres fr this HW set t Lehigh/Clemsn t be psted n Blackbard (I think). I intend t scan the marked up pages and send them t each f yu- but that will take a bit. Questins? Be sure t cntact me at yur cnvenience! rkb