Chapter 3 Chapter 3
Molecular Weight Molecular Weight
1. Thermodynamics of Polymer Solution
2. Mol Wt Determination
1. Weight, shape, and size of polymers 1. Weight, shape, and size of polymers
monomer – oligomer – polymer
dimer, trimer, ---
telomer ~ oligomer from telomerization (popcorn polymerization) telechelic polymer ~ with functional group
macro(mono)mer ~ with polymerizable group [wrong definition p72]
pleistomer ~ mol wt > 1E7
usual range of mol wt of polymers
25000 ~ 1E6
mol wt of chain polymers are higher
molecular weight molecular size
LS (solution) and SANS (bulk) determine size.
conformation Æ shape
2. Solution 2. Solution
ΔG
m= ΔH
m– T ΔS
mΔSm > 0 always
ΔHm > 0 almost always
“like dissolves like”
ΔHm = 0 at best (when solute is the same to solvent) if not, ΔHm > 0
ΔHm < 0 only when specific interaction like H-bonding exists
For solution, ΔHm < T ΔSm
m for mixing
f for melting (fusion)
Solubility parameter Solubility parameter
ΔH
m= V
m[(ΔE
1/V
1)
½– (ΔE
2/V
2)
½]
2v
1v
2= V
m[δ
1– δ
2]
2v
1v
2ΔE ~ cohesive energy ~ energy change for vaporization
ΔE = ΔHvap – PΔV ≈ ΔHvap – RT [J]
ΔE/V ~ cohesive energy density [J/cm3 = MPa]
δ ~ solubility parameter [MPa½]
[MPa½] = [(106 N/m2)½] = [(J/cm3)½] ≈ [(1/2)(cal/cm3)½]
1: solvent
2: solute
Table 3.1 & 3.2
Determination of δ
from ΔHvap data ~ for low mol wt, not for polymers with solvent of known δ
swelling ~ Fig 3.1 viscosity ~ Fig 3.2
group contribution calculation
δ = ρ ΣG / M ~ Table 3.3
G ~ group attraction constant example p79
ΔE = ΔEdispersion + ΔEpolar (+ ΔEHB) δ2 = δdispersion2 + δpolar2 (+ δHB2 )
For solution,
ΔHm < T ΔSm
without specific interaction
δ1 = δ2 at best Æ ΔHm = 0 Æ ΔGm < 0
Δδ < 20 MPa½ (?) ~ for solvent/solvent solution
Δδ < 2 MPa½ ~ a rough guide for solvent/polymer solution
ΔSm smaller
Δδ < 0.1 MPa½ ~ for polymer/polymer solution
semicrystalline polymers not soluble at RT
positive ΔHf Æ ΔHf + ΔHm > T ΔSm
3. Thermodynamics of polymer solution 3. Thermodynamics of polymer solution
Types of solutions
ideal soln ΔHm = 0, ΔSm = – k (N1 ln n1 + N2 ln n2) regular soln ΔHm ≠ 0, ΔSm = – k (N1 ln n1 + N2 ln n2) athermal soln ΔHm = 0, ΔSm ≠ – k (N1 ln n1 + N2 ln n2) real soln
ideal solution
ΔG1 = μ1 – μ1o = RT ln n1 ΔG2 = μ2 – μ2o = RT ln n2
ΔGm = (N1/NA)ΔG1 + (N2/NA)ΔG2
= kT (N1 ln n1 + N2 ln n2)
ΔHm = 0 Æ ΔSm = – k (N1 ln n1 + N2 ln n2)
Eqn (3.9) corrected n: mol fraction
N: number of molecules n1 = N1/(N1+N2)
Eqn (3.12)
Δ Δ S S
mmfrom statistical thermodynamics from statistical thermodynamics
Lattice model
Filling N1 & N2 molecules in N1+N2 = N0 cells
volume of 1 ≈ volume of 2
Boltzmann relation, Sconfigurational = k ln Ω
Ω ~ number of (distinguishable) ways
Ω12 = (N1+N2)!/N1!N2! ΔSm = S12 – S1 – S2
= k ln Ω12 = k ln [(N1+N2)!/N1!N2!]
Ë Sterling’s approximation, ln x! = x ln x – x
ΔSm = k [(N1+N2) ln (N1+N2) – (N1+N2) – N1 ln N1 + ---
= – k (N1 ln n1 + N2 ln n2)
Fig 3.3(a)
S1 = k ln Ω1 = k ln (N1!/N1!) = 0 = S2
Δ Δ S S
mmof polymer of polymer soln soln from stat thermo from stat thermo
developed by Flory & Huggins
polymer soln = mixture of solvent/polymer
volume of 1 << volume of 2 (by x)
A polymer molecule with x mers (repeat units) takes x cells.
volume of 1 mer ≈ volume of 1 solvent molecule
Filling N1 solvents & N2 polymers in N1+ xN2 = N0 cells
ΔSm = S12 – S1 – S2 = k ln [Ω12/Ω1Ω2]
Number of ways to fill the (i+1)th chain in N0 cells νi+1 = (N0-xi) z(1-fi) (z-1)(1-fi) --- (z-1)(1-fi)
1st 2nd 3rd xth segment (mer) z ~ coordination number (# of nearest neighbor) fi ~ probability of a site not available ≈ xi/N0
Fig 3.3(c)
(cont’d)
νi+1 = (N0-xi) z(1-fi) (z-1)(1-fi) --- (z-1)(1-fi)
= (N0–xi) z (z–1)x-2 [1–(xi/N0)]x-1
= (N0–xi) (z–1)x-1 [(N0–xi)/N0]x-1
= (N0–xi)x [(z–1)/N0]x-1
= {(N0–xi)!/[N0–x(i+1)]!} [(z–1)/N0]x-1
(N0-xi)! / [N0-x(i+1)]! =
(N0-xi)(N0-xi-1)(N0-xi-2)----(3)(2)(1) (N0-xi-x)(N0-xi-x-1)---(3)(2)(1)
= (N0-xi)(N0-xi-1)----(N0-xi-x+1)
ΔSm = S12 – S1 – S2 = k ln [Ω12/Ω1Ω2]
Ω12 ~ # of ways to fill N1+N2 molecules in N0 cells
= (1/N2!) Π νi+1 (from i = 0 to N2-1) (x 1)
= (1/N2!) {[N0!/(N0–x)!][(N0–x)!/(N0–2x)!] --- [(N0–(N2–1)x)!/(N0–N2x)!]} [(z–1)/N0]N2(x-1)
= (1/N2!) [N0!/(N0–N2x)!] [(z–1)/N0]N2(x-1)
= [N0!/ N1!N2!] [(z–1)/N0]N2(x-1) << [N0!/ N1!N2!]
Ω1 ~ # of ways to fill N1 solvent molecules in N1 cells = 1 Ω2 ~ # of ways to fill N2 polymer molecules in xN2 cells
~ xN2 mers in xN2 cells ~ Ω2 = 1? No
= (1/N2!) [(xN2)!/(xN2–N2x)!] [(z–1)/xN2]N2(x-1)
= [(xN2)!/N2!] [(z–1)/xN2]N2(x-1)
Allcock p412
Sc = ΔSdis + ΔSm = (a) + (b)
ΔSdis for disorientation ~ equiv to S2 (Ω2) ~ Ω with N1 = 0 ΔSm = Sc – ΔSdis
ΔSm ΔSdis
S = 0 S = Sc = S12
(con’t)
ΔSm = k ln [Ω12/Ω2]
= k ln {[N0!/N1!xN2!] [xN2/N0]N2(x-1)}
Ë Sterling’s approximation, ln x! = x ln x – x
= k {– N1 ln [N1/N0] – N2 ln [xN2/N0]}
= – k [N1 ln v1 + N2 ln v2]
xÇ (mol wt Ç) Æ N2 È Æ ΔSmÈ
for polymer/polymer soln, ΔSm even smaller (N1 & N2 È)
Flory-Huggins theory
volume fraction instead of mole fraction
Eqn (3.16) v ~ volume fraction
Eqn (3.19)
ΔSm = – k [N1 ln v1 + N2 ln v2]
xÇ (mol wt Ç) Æ ΔSmÈ
for polymer/polymer soln, ΔSm even smaller
Examples (for the same v1 = v2 = .5)
case 1: N1=10000, N2=10000, x1 = x2 = 1
ΔSm = – k [10000 ln .5 + 10000 ln .5] = – 20000 k ln .5 case 2: N1=10000, N2=100, x2 = 100; ΔSm = – 10100 k ln .5 case 3: N1=10000, N2=10, x2 = 1000; ΔSm = – 10010 k ln .5 case 4: N1=10, N2=10, x1 = x2 = 1000; ΔSm = – 20 k ln .5
more examples p85
Δ Δ H H
mmof polymer of polymer soln soln
regular solution
ΔHm ≠ 0, ΔSm = – k (N1 ln n1 + N2 ln n2) ΔHm = N1 z n2 Δw
Δw ~ energy change per contact = w12 – [(w11+w22)/2]
for polymer solution
ΔHm = k T N1 v2 χ
χ ~ Flory-Huggins interaction parameter [dimensionless]
kTχ ~ interaction energy (solvent in soln – in pure solvent)
χ È Æ ΔHm È Æ solvent power Ç
ΔHm = Vm [δ1 – δ2]2 v1v2 ~ k T N1 v2 χ Æ χ = β1 + (V1/RT) [δ1 – δ2]2
β1 ~ entropic ≈ 0
χ = χ1 = χ12 Eqn (3.21)
Table 3.4 Eqn (3.28)
1---1
1---2 2---2
See Young pp143-144
Δ Δ G G
mm~ Flory ~ Flory - - Huggins Huggins Eqn Eqn
ΔG
m= ΔH
m– T ΔS
m= kT [N
1ln v
1+ N
2ln v
2+ χN
1v
2]
useful for predicting miscibility (solubility) drawbacks
no volume change self-intersection
for concentrated solutions only (high v2) χ is not purely enthalpic
example calculation p85
Eqn (3.22)
See Young p145
Partial molar free energy of mixing for solvent Partial molar free energy of mixing for solvent
ΔG
1= ∂ΔG
m/∂m
1from Flory-Huggins eqn
ΔGm = kT [N1 ln v1 + N2 ln v2 + χN1v2]
Ë N1 = NAm1, v1 = m1/(m1+xm2), v2 = xm2/(m1+xm2), kNA = R
ΔG1 = RT [ln (1 – v2) + (1 – 1/x)v2 + χv22]
other form of Flory-Huggins eqn
ΔG
1= μ
1– μ
1o= RT ln a
1= RT ln n
1γ
1ΔG
1= μ
1– μ
1o= (μ
1–μ
1o)
ideal+ (μ
1–μ
1o)
xs¾ ideal: (μ1–μ1o)ideal = RT ln n1
¾ excess: (μ1–μ1o)xs = RT ln γ1
m: # of moles
Eqn (3.23)
a: activity
γ: activity coeff.
n: mol fraction
Eqn (A) Sup 2 Young p145-149
Thermo of
Thermo of dilute dilute polymer polymer soln soln
dilute polymer soln
polymer chains separated by solvent FH theory does not hold
In FH theory, chains are placed randomly
Modification ~ Flory-Krigbaum theory
for dil polym soln
n2 = v2/x
É v2 = xN2/(N1+xN2) ≈ xN2/N1 (N1 >> xN2) É n2 = N2/(N1+N2) ≈ N2/N1 (N1 >> N2)
ln v1 = ln (1 – v2) = – v2 – v22/2 – v23/3 – --- ln n = ln (1 – n ) = – n – n 2/2 – n 3/3 – ---
from Eqn (3.23)
ΔG1 = μ1–μ1o = RT [– v2 – v22/2 + v2 + v2/x + χv22]
= –RT(v2/x) + RT(χ – ½)v22 Eqn (3.23-1)
from Eqn (A)
ΔG1 = μ1–μ1o = RT ln n1 + (μ1–μ1o)xs
= –RT(v2/x) + (μ1–μ1o)xs By Flory-Krigbaum
ΔG1xs = (μ1–μ1o)xs = ΔHxs – T ΔSxs
= RTκ v22 – T Rψ v22 = RT(κ – ψ) v22 ΔG1xs = RTψ [(θ/T) – 1] v22 = RT (χ – ½) v22
When T = θ, χ = ½ Æ ΔG1xs = 0 Æ ΔG1= ΔG1ideal
θ-condition (Flory condition) ~ becomes ideal solution
When T > θ, χ < ½ Æ ΔG1xs < 0 Æ soluble
κ = ψθ/T
Table 3.4
–
χ = ½ for ideal
Chapter 3 Chapter 3
Molecular Weight Molecular Weight
1. Thermodynamics of Polymer Solution
2. Mol Wt Determination
4. Mol wt and mol wt distribution 4. Mol wt and mol wt distribution
mol wt distribution
xi ~ number (mol) of i = Ni i ~ molecule having Mi
wi ~ weight (amount) of i = NiMi
Usually xi and wi are fractions xi = Ni/ΣNi , wi = NiMi/ΣNiMi Not in this textbook
Ni xi
Mi
Mi wi
NiMi
mol wt averages mol wt averages
number-average mol wt
(수평균 분자량)weight-average mol wt
(중량평균 분자량)z-average mol wt
viscosity-average mol wt
(점도평균 분자량)= total weight/total number
~ weight of 1 molecule
a dep on solvent & temp
Mn, Mw, Mz are absolute mol wts.
z+1-average mol wt, etc
Eqn (3.31-34)
mol wt distribution (MWD) mol wt distribution (MWD)
Mol wt of polymers almost always has a distribution.
polydisperse (다분산성) ↔ monodisperse (단분산성) polydispersity index (PDI) = Mw/Mn
other indexes; Mz/Mw, Mz+1/Mz …
Most probable distribution (Flory(-Schultz) distribution)
Mn/Mw/Mz = 1/2/3 ideal, not probable practically Mw/Mn > 2 p86 wrong!
Fig 3.4
mol wt & properties mol wt & properties
mol wt independent properties
density, refractive index, solubility, stability, etc dep on repeat unit (chemical) structure
M
ndependent properties
thermal and mechanical properties
Tg, Tm, strength, modulus, etc
dep on segmental motion, chain-end concentration
Tg = Tg∞ – A/Mn Tg
mol wt & properties (2) mol wt & properties (2)
M
wdependent properties
(melt) viscosity
dep on whole chain motion
MWD dependent properties
shear-rate sensitivity of viscosity dep more on larger molecules
log η
log Mw
5. Determination of
5. Determination of M M
nnend-group analyses
step polymers
HOOC---COOH H2N---NH2 HO---OH titration or spectroscopic methods
chain polymers
RMMMM--- (R=initiator fragment) spectroscopic methods
accurate but limited
Colligative property measurements
colligative (collective) property ~ property that depends only on the number of molecules
osmotic pressure, boiling point, freezing point, etc counting number & measuring weight Æ Mn
ΔG1 = μ1 – μ1o = RT ln a1 = RT ln γ1n1 For dilute polymer solution (c2 Æ 0)
Ë solvent behaves ideally, a1 ≈ n1
μ1 – μ1o = RT ln n1 = RT ln (1–n2)
= –RT[n2 + n22/2 + n23/3 + ---]
a: activity
γ: activity coeff.
n: mol fraction c: wt conc’n
n2 = N2/(N1+N2) ≈ N2/N1 = (N2/NA)/(N1/NA)
= m2/m1 = (m2/L)/(m1/L)
= (c2/M2)/(1/V10) [(g/L)/(g/mol)]/[(1/(L/mol)]
= (c2V10)/M2
μ1 – μ1o = –RT[n2 + n22/2 + n23/3 + ----]
= –RTV10[(1/M2)c2 + (V10/2M22)c22 + (V102/3M23)c23 --]
–(μ1 – μ1o)/V10 = RT [(1/M2) c2 + A2 c22 + A3 c23 + ----]
virial equation
A2 ~ 2nd virial coeff, A3 ~ 3rd virial coeff
for dilute polymer soln, c2 Æ 0
[CP/c] = RT/M
colligative property (CP)
CP/c
A2 RT/Mn
c: wt conc’n m: # of moles V10: molar vol M: mol wt
ebulliometry (bp elevation)
ΔTb/c = Ke [(1/Mn) + A2 c + A3 c2 + ----]
Ke calibrated with known mol wt
limited by precision of temperature measurement
useful only for Mn < 30000 not used these days
cryoscopy (fp depression)
ΔTf/c = Kc [(1/Mn) + A2 c + A3 c2 + ----]
Kc calibrated with known mol wt
limited by precision of temperature measurement
useful only for Mn < 30000 not used these days
Eqn (3.35)
Eqn (3.36)
membrane osmometry
static or dynamic method
useful for 30000 < Mn < 10E6
diffusion of solute
h Æ ρgh = π ~ osmotic pressure
μ1(1,P) = μ1(n1,P+π)
μ10(P) = μ10(P) + ∫PP+π V10dP + RT ln a1 πV10 = RTV10 [(1/Mn)c + A2 c2 + A3 c3 ----]
π/c = RT [(1/Mn) + A2 c + A3 c2 + ----]
Eqn (3.41)
Determination without extrapolation?
πV10 = RTV10 [(1/Mn)c+A2 c2+ --] = –RT ln a1 = –(μ1–μ1o) μ1–μ1o = –RT(v2/x) + RT(χ – ½)v22
π = RT(v2/xV10) + RT(χ – ½)v22/V10
Ë v2 ≈ xN2/N1, V = (N1/NA)V10, Mn = ΣNiMi/ΣNi = M2/(N2/NA) Ë c2 = M2/V = MnN2/NAV, ρ2 = V2/Mn, x = V2/V1
π/c = RT(1/Mn) + RT (χ – ½)(1/V1ρ22) c
= RT [1/Mn + A2 c]
At θ-condition, χ = ½ , A2 = 0
no conc’n dependence
determination at 1 conc’n ~ need no extrapolation hard to do ~ not a good solvent (ppt)
Eqn (3.23-1) dil soln
Eqn (3.26)
Fig 3.5
vapor phase (pressure) osmometry (VPO)
P10 –P1 = ΔP
~ vapor pressure drop due to solute
ΔP Æ ΔT Æ Δr
Δr/c = KVPO[(1/Mn) + A2 c ----]
KVPO calibrated with known mol wt at the same temp, drop size, time
Useful for Mn < 30000
small signal (Δr)
6. Determination of M 6. Determination of M
wwlight scattering (LS)
Light scattered by fluctuation in
refractive index (n) Å concentration Å mol wt
Hc/Rθ = 1/Mw + 2 A2c + 3 A3c2 + ---
H = 2π2n02(dn/dc)2/NAλ4
Why Mw? intensity ∝ (amplitude)2 ∝ (mass)2
[Hc/Rθ]cÆ0 = 1/Mw for small molecules, not for polymers
r θ I0
λ
iθ
iθ/I0 = f (dn/dc, M, λ, n0) Rayleigh ratio, Rθ = (iθ/I0)r2
Eqn (3.43)
for large molecules (D > λ/20)
Hc/Rθ = 1/(MwP(θ)) + 2 A2c + ---
P(θ) = scattering (form) factor = Rθ/R0 1/P(θ) = 1 + (8π2/9λ2)<r2>sin2(θ/2)
= 1 + (16π2/3λ2)<Rg2>sin2(θ/2)
r = end-to-end distance Rg = radius of gyration
<r2>0 = 6 <Rg2>0
Hc/Rθ = 1/Mw + (16π2/3λ2Mw)<Rg2>sin2(θ/2) + 2 A2c + ---
[Hc/Rθ]θ=0 = 1/Mw + 2 A2c + ---
[Hc/Rθ]c=0 = 1/Mw + (16π2/3λ2Mw)<Rg2>sin2(θ/2)
r
Rg
Eqn (3.50), Fig 3.10(c)(d)
i30 ≠ i45
Eqn (3.51), Fig 3.10(a)(b) Eqn (3.61)
7. MW of common polymers 7. MW of common polymers
MW of commercial polymers
step polymers: 20000 – 40000
chain polymers: 20000 – 1000000 MWD
Flory-Schultz distribution: PDI = 2
when ideal
Poisson distribution: PDI = 1
anionic living polymerization
In most polymerizations: PDI > 2
Table 3.9
PDI(chain polymers) > PDI(step polymers)
8. Determination of
8. Determination of M M
vvdilute solution viscometry (DSV)
viscosity Å size mol wt measures molecular size, not weight
not an absolute method, but a relative method
viscosity, η
η = η0 (1 + ωv2) Einstein eqn
ω = 2.5 for sphere v2 ∝ size of solute
η/η0 – 1 = 2.5 N2Ve/V
ηrel – 1 = ηsp = 2.5 cNAVe/M (g/L)(1/mol)(L)/(g/mol)
[η /c] = 2.5 N V /M = [η] ‘intrinsic viscosity’
η0: solvent only
v2: vol fraction of solute V: vol of soln
Ve: vol of equiv. sphere c: wt conc’n
M: mol wt
for η’s, see handout p92 Fig 3.13
Fig 3.12
shape
[η] Æ mol wt
Ve = (4/3)πRe3 = (4/3)πH3Rg3 ∝ (4/3)πH3(M/M0)3/2
É Re = HRg, Rg ∝ (M/M0)½ at θ-condition
[η]θ = 2.5 NAVe/M ∝ (10πNAH3/3M03/2)M½ [η]θ = Kθ M0.5 at θ-condition
in good solvent, Rg = α Rgθ
[η] = α3 [η]θ = α3 Kθ M0.5 = λ3 Kθ M(0.5+3Δ) = K Ma
É α = λ MΔ
[η] = K Mva ‘Mark-Houwink-Sakurada (MHS) eqn’
Mv~ viscosity-average mol wt a ≥ 0.5
M
v~ viscosity-average mol wt
ηsp = Σ(ηsp)i ~ Ni moles of Mi mol wt
= Σ ci [η]i = Σ (NiMi) (KMia) = K ΣNiMi1+a [η] = [ηsp/c]cÆ0 = K ΣNiMi1+a/ΣNiMi = K Mva Mv = [ΣNiMi1+a/ΣNiMi]1/a
0.8 ≥ a ≥ 0.5
0.5 at θ-condition
when a = 1, Mv = Mw when a = –1, Mv = Mn Mv close to Mw
DSV experiment
capillary viscometer Å Poiseulli eqn, Q = V/t = πr4P/8ηL η ∝ t Æ η/η0 = t/t0
Procedure
measure t0, t1, t2 --- at c0, c1, c2 --- (0 ~ solvent) [η] = [ηred]cÆ0 = [ηsp/c]cÆ0 = [(ηrel – 1)/c]cÆ0
= [(η/η0 – 1)/c]cÆ0 = [(t/t0 – 1)/c]cÆ0 or [η] = [ηinh]cÆ0 = [ln ηrel/c]cÆ0
= [ln (η/η0)/c]cÆ0 = [ln (t/t0)/c]cÆ0 [η] = K Mva
K, a from handbook at the same temp and solvent
Cautions: temp control < 0.2 K t0 > 100 s (laminar)
c < 1 g/dL (Newtonian)
Table 3.10 Fig 3.14
9. Gel Permeation Chromatography (GPC) 9. Gel Permeation Chromatography (GPC)
size exclusion chromatography
separation by size using porous gel substrate Larger molecules elute earlier.
Instrumentation
injector – column(s) – detector
A chromatogram
Fig 3.16-18
VR: retention volume VR = tR x flow rate
VR ~ size Æ mol wt (Mi) Hi ~ amount Æ NiMi
Needs calibration
Universal calibration
With the same instrument, column, and solvent, the same VR represents the same hydrodynamic volume.
[η] M = [2.5 NA V] = K Mva+1
Many polymers fall on the same curve on the [η]M – VR plot ~ universal calibration curve
Procedure of an experiment
1. From the chromatogram, read VRi and Hi
(column 1 and 2).
2. Run the same experiment with polystyrene standards.
Fig 3.23
2 (cont’d) PS standard anionically polymerized with known mol wt
3. Draw a calibration curve (Mi vs VRi) for PS.
4. Read Mi (PS) for each VRi (column 1 , sample).
5. M (PS) Æ M (sample)
[η]PSMPS = [η]sampleMsample
KPSMPSa(PS)+1 = KsampleMsamplea(sample)+1
6. Ni = Hi / Mi (column 2 / column 3) 7. Calculate Mn, Mw, MWD.
Are Mn and Mw obtained absolute? No.
from handbook
from calibration curve column 3
10. Mass spectrometry 10. Mass spectrometry
MS determines mol wt
by detecting molecular ion, M+ in vapor phase
ionization of polymers in gas phase?
MALDI-TOF technique
soft ionization
choice of the matrix critical for not-too-high mol wt
useful for (highly) branched polymers
11. Conclusions 11. Conclusions
At theta condition (solvent/temp)
A2 = 0, χ = ½
Rg is the same to that of the bulk polymer
infinite mol wt fraction just precipitate (poor/good)
Absolute and relative methods
Table 3.15