Dynamic Surface Tension Measurement with a Dynamic Wilhelmy Plate Technique

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Journal of Colloid and Interface Science215,258–269 (1999)Article ID jcis.1999.6270, available online at http://www.idealibrary.com on

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Dynamic Surface Tension Measurement with a DynamicWilhelmy Plate Technique

Ning Wu,* Jialin Dai,† and Fortunato J. Micale‡

*Sun Chemical Corporation, 631 Central Avenue, Carlstadt, New Jersey 07072;†Department of Chemical Engineering,Southwest Petroleum Institute, Nanchong, Sichun 637001, People’s Republic of China; and‡Zettlemoyer Center

for Surface Studies, Lehigh University, Sinclair Lab 7, Bethlehem, Pennsylvania 18015

Received July 28, 1998; accepted April 14, 1999

rta–3ndthean

el iatThth

he. 1rtins

I (t erex kc er-e l (4):

wthe

f ion.A ben termb

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An experimental method called dynamic Wilhelmy plate tech-ique (DWPT) for studying dynamic surface tension was designed

n this study. A diffusion-controlled model corresponding to thenitial and boundary conditions of this method was proposed.ynamic surface tension of Triton X-100 and SDS was measuredith this technique and analyzed with the proposed model. The

alculated diffusion coefficients for the short- and long-time ap-roximations are 3.7 3 1026 and 0.97 3 1026 cm2/s for Triton-100 and 4.6 3 1026 and 0.79 3 1026 cm2/s for SDS, respectively.he predicted dynamic surface tension with these diffusion coef-cients for the simultaneously generated aqueous/air interfaces is

n good agreement with the drop mass technique. Another diffu-ion controlled model that considers the energy barrier at thequeous/air interface was also proposed in this study. The calcu-ated energy barriers are in the range of 4.1–5.7 RT for Triton-100 and 6.5–8.0 RT for SDS. © 1999 Academic Press

Key Words: dynamic surface tension; dynamic Wilhelmy plateechnique; short-time approximation; long-time approximation;iffusion coefficient.

INTRODUCTION

The studies in dynamic surface tension are very imporom both the theoretical and the practical points of view (1he diffusion controlled model, initially proposed by Ward aordai in 1946 (4) and further developed by several oesearchers (5–9), has been the generally accepted mechor dynamic surface tension. The diffusion controlled modased on the assumption that the surfactant adsorptionqueous/air interface is a diffusion controlled process.

ime dependence of surface tension is directly related tourface excess (G) at the surface, which is a function of tiffusion rate of surfactant molecules to the interface (Fighe diffusion controlled model can be expressed by a paifferential equation with the boundary and initial conditioefined as follows (10–12).Fick’s second law of diffusion:

c

t5 D

2c

x2 ~ x . 0, t . 0!. [1a]

258021-9797/99 $30.00opyright © 1999 by Academic Pressll rights of reproduction in any form reserved.

nt).

rism

stheee

).al

Boundary conditions:

G

t5 D

c

x~ x 5 0, t . 0! [1b]

c 5 c0 ~ x3 1`, t . 0!. [1c]

Initial conditions:

G 5 0 ~t 5 0! [1d]

c 5 c0 ~ x $ 0, t 5 0!. [1e]

n the above equations,c is the surfactant concentration atx,), t is the time,x is the distance from the subsurface wh5 0, D is the diffusion coefficient,c0 is the surfactant bul

oncentration, andG is the surface excess. This partial diffntial equation can be expressed by the convolution integra

G 5 2c0ÎDt

p2 2ÎD

p E0

t 1/ 2

csub~t 2 f!df 1/ 2, [2]

heref is a dummy time parameter.The first term on the right-hand side of Eq. [2] is for

orward diffusion and the second term is for the back diffust the early stage of adsorption, the back diffusion caneglected due to the fairly empty surface, so the firstecomes the short-time approximation:

G 5 2c0 ÎDt

p. [3]

Jooset al. (5) derived a long-time approximation by assung a constant subsurface concentration,

g t 5 ge 1RTG e

2

c Î p

4Dt, [4]

0

we x-cc -p

(212 eed nsis ndt Bc ths agt tems mo req s od osid laa

P

set tioa (F2 wid tiog n b

r stantc rfac-t aterl us/airi timeu ledp froms nceb ctants (thes ctivei terl tionp rface,b ningt idss le toa cos-i

E

“dy-n int ump 0 areu emati-c ock.

aterl te isp at afi thes thew e Fig.5 on oft nt ad-s lmyp y theC thed

eter

ousa atea tratb ntss ntrag adso stat .

259DYNAMIC SURFACE TENSION

hereg t is the dynamic surface tension at timet, ge is thequilibrium surface tension,Ge is the equilibrium surface eess,c0 is the surfactant bulk concentration,D is the diffusionoefficient,R is the gas constant, andT is the absolute temerature.The drop volume (17–19), maximum bubble pressure

3), oscillating jet (24), and dynamic capillary (25) have beveloped and extensively used in dynamic surface tetudies. These techniques involve the surface expanding ahe solution flow creating the new aqueous/air interface.ause the viscous force plays an important role duringurface expansion, especially at relatively smaller surfacehese techniques are limited for the higher-viscosity sysuch as polymer–surfactant mixed solutions. The measuref dynamic surface tension for higher-viscosity solutionsuires either complicated calibration of current techniqueesign of a new experimental technique where the viscoes not cause interference in the measurement. Thepproach was adopted in this research.

EXPERIMENTAL

rinciple

Based on the diffusion controlled mechanism, it is propohat if a water layer is introduced above the surfactant solund they are initially separated by a nonpermeable barrier), by taking away the barrier, the surfactant moleculesiffuse into the water layer and results in a concentraradient. The surfactant solution below the water layer ca

FIG. 1. Diffusion controlled mechanism for a newly generated aqueir interface proposed by Ward and Tordai (4). 1, Initial state, immedifter the generation of a new aqueous/air interface. Surfactant concenelow the surface is uniform. 2, 3, and 4, Intermediate stages, surfactaubsurface adsorb onto the aqueous/air interface and result in a conceradient below the surface, and the surfactants continue to diffuse andnto the aqueous/air interface under the concentration gradient. 5, Final

he surfactant concentration below the surface is uniform at equilibrium

–non/or

e-ee,s

ent-r

tytter

dn,ig.llne

egarded as an infinite surfactant reservoir with a cononcentration if the dimensions and the volume of the suant solution are significantly larger than the introduced wayer. When the surfactant molecules reach the aqueonterface, the surface tension will decrease as a function ofntil the equilibrium is reached. This is a diffusion controlrocess if we assume that the adsorption of surfactantsubsurface to surface is diffusion controlled. The differeetween the process with a water layer above the surfaolution (Fig. 2) and the one without the water layerimultaneously generated surface, Fig. 1) is their respenitial and boundary conditions. The introduction of a waayer above the surfactant solution will delay the adsorprocess compared to that seen with the new generated suut makes it possible to obtain more information concer

he early stage of adsorption. Most importantly, it avourface expansion and solution flow; therefore it is possibpply it to study the dynamic surface tension of higher-vis

ty systems.

xperimental Apparatus

Based on the above considerations, a technique calledamic Wilhelmy plate technique” (DWPT) was designed

his lab, where a large-bore stopcock, a Wilhelmy platinlate, and a null reading and recording Cahn Balance 100sed to set up the experimental apparatus as shown schally in Fig. 3. Figure 4 shows a cross section of the stopcThe stopcock contains the surfactant solution, and the w

ayer is introduced into the upper cup. The platinum plaartially immersed into the water layer, where it remainsxed position during the course of the experiment. Whentopcock is opened the surfactants can diffuse throughater layer and adsorb onto the aqueous/air interface (se), and the surface tension will thus decrease as a functi

ime. The surface tension decrease due to the surfactaorption results in the force changes applied on the Wilhelate. The force changes are continuously measured bahn balance and recorded by the recorder, from whichynamic surface tension can be calculated.The inside volume of the stopcock is 83.3 ml. The diam

/lyionat

tionorbge,

FIG. 2. The basic consideration for the experimental design.

a cmr con 0 mo lcul Thl ar1

anw bs k.

ity1 tm naw omp lsc thf sf ac n at emf .5%

w me-c

C

sedi [5],a ue(

w y the

ue.( eo

260 WU, DAI, AND MICALE

nd length of the stopcock are approximately 4.6 and 5.0espectively. The diameter and depth of the upper cupected to the stopcock are 1.60 and 4.0 cm, into which 2.0f water was introduced with a volumetric pipet. The ca

ated height of this introduced water layer is 1.00 cm.ength, width, and thickness of the platinum Wilhelmy plate0, 2.9, and 0.06 mm, respectively.After loading the surfactant solution in the stopcockater in the upper cup, the experiment can be initiatedlowly (typically 2 min) and carefully opening the stopcocThe Cahn balance (Cahn Company, CA) has a sensitivmg and can measure a maximum force of 10g. The weigheasured by the Cahn Balance is converted to voltage sighich are sent to the strip chart recorder (The Recorder Cany, TX). The recorder traces continuous voltage signaontrolled paper speed of 1 cm/s to 1 cm/h, which permitsorce applied on the Wilhelmy plate to be measured aunction of time. The stopcock and the Cahn balanceontained in a closed system to reduce water evaporatiohe disturbance caused by the airflow. Repeated measuror the same sample show that the variation is about 0

FIG. 3. The dynamic Wilhelmy plate technique (drawn withou

FIG. 4. The cross section of the stopcock when it is closed.

,n-

l-ee

dy

of

ls,-

atea

rendent,

hich might result from small temperature variation andhanical disturbance during the experiments.

alculation of the Dynamic Surface Tension

The force applied on the Wilhelmy plate partially immernto a liquid can be related to the surface tension by Eq.ccording to the principle of the Wilhelmy plate techniq24),

F 5 pg cosu 2 rgV, [5]

hereF is the force applied on the plate and measured b

mensions) (1, Surfactant solution; 2, Solvent layer; 3, Wilhelmy plate).

FIG. 5. The physical picture for the dynamic Wilhelmy plate techniqa) t 5 0, immediately after opening the stopcock; (b)t 5 t, during the coursf the experiment; (c)t 3 `, at equilibrium.

t di

C hel heW nd tei r os cab f E[ ndt

canb

w ena

bew

w luec

ena

w est

thp hot the

r forcem

E

s ofT ea-s her d asr calm alcu-l ten-s 1.

s ofT 025,a and0 ratusd latedb

T

thea micW e ther news usionc rface(

sionc ed,ab aterl0 n thes wasm ck iss sur-f s ani port

a

TS


ahn balance,p is the perimeter of the Wilhelmy plate at tiquid/air interface,u is the contact angle of the liquid on t

ilhelmy plate,r is the liquid density,g is the acceleratioue to gravity, andV is the volume of the part of the pla

mmersed in the liquid. The contact angle of the wateurfactant solutions on the Platinum plate is very small ande regarded as zero. The first term on the right-hand side o

5] is the force due to the surface tension, while the secohe force due to the buoyancy effect.

At time t 5 t 0, e.g., the opening of the stopcock, Eq. [5]e written as

F0 5 pg0 2 rgV, [6]

hereF 0 is the force applied to the plate by the pure solvndg0 is the surface tension of pure solvent.After reaching equilibrium, a similar relationship canritten as

Fe 5 pge 2 rgV, [7]

here the terms in this equation are the equilibrium vaorresponding to Eq. [6].At any time t 5 t during the experiment, it can be writt

s

Ft 5 pg t 2 rgV, [8]

here the terms in this equation correspond to the valuime t.

The perimeter of the plate at the liquid/air interface andlate volume immersed into the liquid are constant throug

he experiment. Combination of Eqs. [6]–[8] results in

FIG. 6. Equilibrium surface tension of Triton X-100 (h) and SDS (‚) asfunction of concentration in aqueous solutions.

rnq.is

t,

s

at

eut

elationship between the dynamic surface tension and theeasured at timet:

g t 5 g0 2~F0 2 Ft!~g0 2 ge!

~F0 2 Fe!. [9]

xperimental

The equilibrium surface tension of the aqueous solutionriton X-100 (Rame and Haas) and SDS (Kodak) was mured with du Nou¨y ring method on the Cahn balance. Tesults are shown in Fig. 6. The surfactants were useeceived without further purification. The values of critiicelle concentration (cmc), saturated surface excess c

ated with the Gibbs adsorption equation and the surfaceion of the surfactant solutions at cmc are listed in TableThe forces applied on the Wilhelmy plate for the solution

riton X-100 at the concentrations of 0.003, 0.0006, 0.00nd 0.0001 M and SDS at concentrations of 0.05, 0.01,.005 M were measured as functions of time with the appaescribed above. The dynamic surface tension was calcuy Eq. [9] and the results were shown in Figs. 7 and 8.

THEORETICAL MODELS

he Partial Differential Equation for DWPT

A specific diffusion controlled model corresponding toctual initial and boundary conditions involved in the dynailhelmy plate technique was proposed in order to analyz

esults and apply them to the instantaneously generatedurface. The model proposed here is an analog to the diffontrolled model for a newly generated aqueous/air inteEqs. [1]–[4]).

The following assumptions were made to derive a diffuontrolled model for DWPT: (i) When the stopcock is opensharp boundary is formed atx 5 0, i.e., the dividing line

etween the surfactant solution in the stopcock and the wayer in the upper cup. (ii) The surfactant concentration atx 5

is constant and equal to the surfactant concentration itopcock throughout the experiments. This assumptionade because the volume and dimension of the stopco

ignificantly larger than the introduced water layer. Theactant solution inside the stopcock can be regarded anfinite surfactant reservoir (26). (iii) The surfactant trans

TABLE 1Calculated Constants from the Equilibrium

Surface Tension Measurement

Saturated surface excess,Gm (mol/cm2)

CMC(M)

gCMC

(dyn/cm)

riton X-100 3.093 10210 3.03 1024 30.1DS 6.273 10210 4.83 1023 36.8

i oua icap n ta

ntie ndi atet

I atd c-t thes eb , ax

hat

d aterl ryc f thee ningb

art e tot t isd 0f]o uirest orp-t tters

S

theia

w

w0c],

a enh

ofcM

ationmE


n bulk and adsorption from the subsurface onto the aqueir interface is a diffusion controlled process. The physictures for the diffusion and adsorption process based obove assumptions is shown in Figs. 5a–5c.Based on the assumptions above, the partial differe

quation with the initial and boundary conditions correspong to the physical situation of the dynamic Wilhelmy plechnique can be expressed as follows.

Fick’s second law of diffusion:

c

t5 D

2c

x2 ~ x . 0, t . 0!. [10a]


G

t5 D

c

x~ x 5 h, t . 0! [10b]

c 5 c0 ~ x 5 0, t . 0!. [10c]

Initial conditions:

G 5 0 ~t 5 0! [10d]

c 5 c0 ~ x 5 0, t 5 0! [10e]

c 5 0 ~h . x . 0, t 5 0!. [10f]

n the above equations,c is the surfactant concentrationistancex and timet, D is the diffusion coefficient of surfa

ant molecules,c0 is the surfactant concentration insidetopcock,G is the surface excess,x 5 0 corresponds to thoundary between the surfactant solution and water layer5 h corresponds to the aqueous/air interface.Equation [10a] is the Fick’s second law of diffusion t

FIG. 7. Dynamic surface tension for Triton X-100 as a functiononcentration measured with the dynamic Wilhelmy plate technique (h, 0.003; ‚, 0.0006 M;E, 0.00025 M;3, 0.0001 M).

s/l

he

al-

nd

efines the diffusion of the surfactant molecules in the wayer (h . x . 0). Equation [10b] defines the boundaondition at the aqueous/air interface during the course oxperiments. Equations [10c] to [10f] describe the remaioundary and the initial conditions.Comparing Eqs. [10a]–[10f] with Eqs. [1a]–[1e], it is cle

hat the boundary and initial conditions have changed duhe introduced water layer. Similar to Eqs. [1a]–[1e], iifficult to obtain the analytical solution for Eqs. [10a]–[1ver the entire time frame. Mathematical necessity req

hat the short-time approximation for the early stage of adsion process and the long-time approximation for the latage of the adsorption process be treated separately.

hort Time Approximation for DWPT

The Laplace Transformation of Eq. [10a] subjected tonitial condition (att 5 0, c (h . x . 0) 5 0) can be writtens

pc# 5 D 2c#

x2 , [11]

herec# is the form of the Laplace Transformation ofc.The solution of Eq. [11] is given as

c# 5 A1expS Îp

DxD 1 A2expS2Îp

DxD , [12]

hereA1 andA2 are constants.According to the boundary condition imposed by Eq. [1

t x 5 0, c 5 c0 at any time during the experiments. We thave

A1 1 A2 5 c0/p. [13]

FIG. 8. Dynamic surface tension for SDS as a function of concentreasured with the dynamic Wilhelmy plate technique (h, 0.05 M;‚, 0.01 M;, 0.005 M).

WT

a

thw erit roms bsf

onc culw atea f tha E[

tima

suf rfae yf

T facet ivenb

ws et top-cc era [21]i g an

L

rstl

w

ceT sub-s

c

imea iso-t uilib-r e sur-f


henx3 1`, the concentration should be finite, soA1 5 0.hus Eq. [12] can be written as

c# 5c0

pexpS2Îp

DxD . [14]

The differentiation of Eq. [14] results in

c#

x5 2

c0

ÎDpexpS2Îp

DxD . [15]

The inverse Laplace transformation of Eq. [15] is given

c

x5 2

c0

ÎpDtexpS2

x2

4DtD . [16]

Equation [16] gives the concentration gradient insideater layer. The negative sign can be taken away if consid

he concentration gradient by the other direction, i.e., furface to bulk. Thus, the concentration gradient at the suace wherex 5 h is

S c

xDx5h

5c0

ÎpDtexpS2

h2

4DtD . [17]

Since the surface excess for the short time approximatilose to zero, it is assumed that all of the surfactant molehich arrive at the subsurface will be adsorbed onto the wir interface. Thus the back diffusion at the early stage odsorption process can be neglected. Then combining

10b] and [17] gives the rate of adsorption as

SG

t Dx5h

5 DS c

xDx5h

5 c0ÎD

ptexpS2

h2

4DtD . [18]

The surface excess as a function of time for the short-pproximation can be obtained by integrating Eq. [18]:

G~t! 5 c0ÎD

p E0

t

t 21/ 2expS2h2

4DtDdt. [19]

If assuming that the Langmuir relationship defines theace tension as a function of the surface excess, the suxcess and dynamic surface tension can be correlated b

ollowing integrated Gibbs adsorption equation (27):

g0 2 ge 5 GmlnS1 2Ge

GmD . [20]

s

eng

ur-

isesr/e

qs.

e

r-cethe

he result for the short-time approximation of dynamic surension for the dynamic Wilhelmy plate technique is thus gy

g~t! 5 g0 1 RTGmlnF1 2c0

GmÎD

p E0

t

t 21/ 2expS2h2

4DtDdtG ,

[21]

hereg(t) is the dynamic surface tension at timet, g 0 is theolvent surface tension,R is the gas constant,T is the absolutemperature,c0 is the surfactant concentration inside the sock, Gm is the saturated surface excess,D is the diffusionoefficient, andh is the height of the introduced water laybove the surfactant solution. The integration term in Eq.

s difficult to be integrated analytically and requires usinumerical technique.

ong Time Approximation for DWPT

The Laplace Transformation of Eq. [10b] (the Fick’s fiaw of diffusion) can be written as

pG# 5 DS c#

xDx5h

, [22]

here theG# is the form of Laplace Transformation ofG.At x 5 h, the differentiation of Eq. [14] is given as

Sc#

xDx5h

5 2A1Îp

DexpS2h Îp

DD1 A2 Îp

DexpSh Îp

DD .

[23]

Combining Eqs. [12], [13], [22], and [23], the Laplaransformation which correlates the surface excess andurface concentration is given as

# 5

2c0

p

expSh Îp

DD 1 expS2h Îp

DD

1

expSh Îp

DD 2 expS2h Îp

DDexpSh Îp

DD 1 expS2h Îp

DD G# [24]

The back diffusion plays an important role for the long-tpproximation and cannot be neglected. An adsorption

herm must also be considered by assuming that quasi-eqium exists between the subsurface concentration and thace excess:

inp theL anc hodi

r an roxm sE trod cht y, tr

e tt n.i tid thd thw hus ionf tf forD

T

ic croa rfa

w ndQ

thee

andE sur-f Eqs.[

D

D

Ic sdG

w


G 5csub

csub1 aGm. [25]

Though combining Eq. [25] with Eq. [24] should lead,rinciple, to the long-time approximation, the inversion ofaplace transformation is very mathematically involvedannot be solved analytically. The following empirical mets therefore adopted in this study.

Comparing Eq. [3] (the short-time approximation foewly generated interface) and Eq. [19] (short-time appation for the DWPT), it was found that the surface excesq. [19] is greatly delayed. This delay results from the inuced water layer. Ift 1 andt 2 are defined as the time to rea

he same surface excess for Eqs. [3] and [19], respectivelelationship between the timet 1 and t 2 can thus be given as

t1 51

4 F E0

t2

t 21/ 2expS2h2

4DtDdtG 2

. [26]

Equation [26] quantitatively describes the time delay duhe introduced water layer for the short-time approximatios reasonable to assume that the same relationship of theelay also applies to the long-time approximation, sinceelay for the long-time approximation also originates fromater layer introduced above the surfactant solution. Tubstituting Eq. [26] into Eq. [4] (the long-time approximator the new generated aqueous/air interface) results inollowing empirical equation of long-time approximationWPT:

g t 5 ge 1RTG e

2

c0 * 0t t 21/ 2expS2

h2

4DtDdtÎp

D. [27]

heoretical Model Considering the Energy Barrierat Aqueous/Air Interface

For long-time approximation, the surface adsorptionlosed to saturation and surfactant molecules may have ton energy barrier to be adsorbed onto the aqueous/air inte

E 5 expS2Q

RTD , [28]

hereR is the gas constant,T is the absolute temperature, ais the energy barrier term.The rewriting of Eqs. [1a]–[1e] with the consideration of

nergy barrier is presented below.

d

i-in-

he

oItmeees,

he

sss

ce,

Fick’s second law of diffusion:

c

t5 D

2c

x2 ~ x . 0, t . 0!. [29a]


G

t5 E z D

c

x~ x 5 0, t . 0! [29b]

c 5 c0 ~ x3 1`, t . 0!. [29c]

Initial conditions:

c 5 c0 ~ x . 0, t 5 0! [29d]

G 5 0 ~t 5 0!. [29e]

Equation [29a] describes the surfactant bulk diffusionq. [29b] describes the surfactant diffusion from the sub

ace region to the surface. The dimensionless form of29a]–[29e] can be written as follows.

Dimensionless Fick’s second law of diffusion:

C

u5

2C

X2 ~X . 0, u . 0!. [30a]

imensionless boundary conditions:

j

u5 E

C

X~X 5 0, u . 0! [30b]

C 5 1 ~X3 1`, u . 0!. [30c]

imensionless initial conditions:

j 5 0 ~u 5 0! [30d]

C 5 1 ~X . 0, u 5 0!. [30e]

n the above equations, the dimensionless concentrationC 5/c0, dimensionless surface excessj 5 G/G0, dimensionlesistanceX 5 x/k, dimensionless timeu 5 D t/k2, andk 5

0/c0.The Laplace transformation of Eq. [30a] is given as

C# ~X, p! 51

p1 A exp~2ÎpX!, [31]

hereA is a constant to be determined.

heL bsuf

ts . Td

wlo

t t

cen-t iso

s in

r then aque-o howst rptionp

6],r thee

w thee


At X 5 0, Eq. [31] becomes

C# ~0, p! 51

p1 A. [32]

Differentiation of Eq. [31] gives

C# ~X, p!

X5 2ÎpA exp~2ÎpX!. [33]

At X 5 0, Eq. [33] becomes

C# ~0, p!

X5 2ÎpA. [34]

The Lapalace transformation of Eq. [30b] is given as

pj# 5 EC# ~0, p!

X. [35]

Combination of Eqs. [32], [34], and [35] results in taplace transformation of the relationship between the su

ace concentration and the surface excess:

C# ~0, p! 51

p2

Îp

Ej# . [36]

It is assumed that the subsurface concentration andurface excess obey the Langmuir adsorption isothermimensionless Langmuir isotherm can be written as

j 5bC 1 C

bC 1 1, [37]

hereb is equal toc0/a anda is the Langmuir constant.For the long-time approximation, the surface excess is c

o the equilibrium surface excess. It can be assumed tha

j 5 1 1dj

dC~C 5 1!~C 2 1!. [38]

The combination of Eqs. [37] and [38] results in

j 5 1 1C 2 1

1 1 b. [39]

The Laplace transformation of Eq. [39] is given as

j# 5b

~1 1 b! p1

C#

1 1 b. [40]

r-

hehe

se

Combination of Eqs. [36] and [40] gives

C# ~0, p! 5E~1 1 b!

p@E~1 1 b! 1 Îp#2

b

Îp@E~1 1 b! 1 Îp#.

[41]

The inverse Laplace transformation of Eq. [41] is

C 5 1 2 ~1 1 b!exp@E2~1 1 b! 2u#erf@E~1 1 b!Îu#.

[42]

Expanding Eq. [42], the relationship of subsurface conration as a function of time in the dimensionless formbtained as

C~0, u ! 5 1 21

EÎpu. [43]

Substitution of the dimensional terms into Eq. [43] result

Dc~0, t! 5 c~0, t! 2 c0 5 2Ge

EÎpDt. [44]

The Gibbs adsorption equation can be written as

dg 5 2RTG~c!dc

c. [45]

Substitution of Eq. [44] into Eq. [45] results in

g t 5 ge 1RTG e

2

Ec0Î 1

pDt. [46]

Equation [46] represents the long-time approximation foewly generated surface when the energy barrier at theus/air interface are taken into account. This equation s

hat the dynamic surface tension near the end of the adsorocess is inversely proportional to the termE.Substitution of the delay term (Eq. [26]) into Eq. [4

esults in the long-time approximation which considersnergy barrier for the DWPT,

g t 5 ge 12RTG e

2

Ec0 * 0t t 21/ 2expS2

h2

4DtDdtÎ 1

pD, [47]

here the diffusion coefficient can be calculated fromxperimental data of the short-time approximation.Equation [47] can be written as

g 5 g 1 Kf~t!, [48]
t e

w

a

eed anb

e dn tioa her e-c bym w

1 iums thatf utest ssict ffused ch-n ctants h thew us/airi thea latet d cal-i dur-i , andm canb igherv flowb

ctantc sion,w Theh ses ag lts in

entc tra-

t


here

f~t! 5ÎD

* 0t t 21/ 2expS2

h2

4DtDdt

[49]

nd

K 52RTG e

2

Ec0DÎp. [50]

Equation [48] indicates that a linear relationship betwynamic surface tension andf(t) exists. The energy barrier ce calculated from the slopeK.

RESULTS AND DISCUSSION

The dynamic surface tension results measured with thamic Wilhelmy plate technique as a function of concentrare shown in Fig. 7 for Triton X-100 and Fig. 8 for SDS. Tesults show that 1 to 3 h isrequired for the surfactant molules to reach the aqueous/air interface, as indicatedeasurable surface tension change. The results also sho

FIG. 9. The short-time approximation for the Triton X-100 at differoncentrations (symbols as in Fig. 7).

TABThe Calculated Diffusion Coefficients for

TypeConcentration

(M)D short

(106 cm2/s)

Triton X-100 0.003 4.00.0006 3.20.00025 3.70.0001 3.7

SDS 0.05 4.20.01 4.90.005 4.5

n

y-n

athat

0 to 40 h is required for the solutions to reach the equilibrurface tension. The time scale is significantly longer thanor a classic technique, which requires only seconds to mino reach the equilibrium. The physical situation of the claechniques allows the surfactants below the surface to diirectly onto the surface. The dynamic Wilhelmy plate teique, however, introduces a water layer above the surfaolution and the surfactant molecules must diffuse througater layer before reaching and adsorbing onto the aqueo

nterface. This water layer therefore significantly delaysdsorption process. However, the dynamic Wilhelmy p

echnique requires more straightforward data analysis anbration procedures than the classic techniques. The timeng the experiment is the real surface age at that instance

ost importantly, the dynamic Wilhelmy plate techniquee used for the dynamic surface tension measurement of hiscosity systems because of the absence of solutionrought about by an expanding surface.Figures 7 and 8 also show that the higher the surfa

oncentration, the faster the drop in dynamic surface tenhich is similar to that seen in the classic techniques.igher surfactant concentration inside the stopcock caureater concentration gradient in the water layer and resu

FIG. 10. The short-time approximation for SDS at different concenions (symbols as in Fig. 8).

2e Short- and Long-Time Approximation

D long

(106 cm2/s)D short (ave.)(106 cm2/s)

D long (ave.)(106 cm2/s)

1.10.970.950.92 3.7 0.970.770.810.79 4.6 0.79

LEth

a ams

ionw rard ths dy-n the ano cmC od ai ee iffs

tiona DSr nT entc e at s t

d olec-u chh hant eser erser ularw

ionc -timea sti-t en-t lts oft igs.1 ares

on-c

ion(

line,D withd0

line,D withd


faster diffusion and adsorption rate; therefore, the dynurface tension drops faster.The diffusion coefficients for the short-time approximatere obtained according to the following procedure: Arbitiffusion coefficients were substituted into the model forhort-time approximation (Eq. [21]) until the predictedamic surface tension curve fits the initial decrease ofxperimentally measured dynamic surface tension. The rf this initial decrease should not be over 1 or 2 dyn/alculation proves that 2 dyn/cm of the initial decreaseynamic surface tension for Triton X-100 corresponds to

ncrease in surface excess of 0.23Gm. This amount of surfacxcess at the aqueous/air interface is sufficient for back dion to be a significant factor in the adsorption process.The results of the curve fitting for short-time approxima

re shown in Figs. 9 and 10 for Triton X-100 and Sespectively. The obtained diffusion coefficients are showable 2. The variation of diffusion coefficients at differoncentrations may result from the mechanical disturbancemperature variation during the experiment. Table 2 show

FIG. 11. The long-time approximation for Triton X-100 at various centrations (symbols as in Fig. 7).

FIG. 12. The long-time approximation for SDS at various concentratsymbols as in Fig. 8).

ic

ye

ege.fn

u-

,in

ndhe

ependence of diffusion coefficients on the surfactant mlar weight. The diffusion coefficient for Triton X-100, whias a molecular weight of 624.8, is significantly smaller t

hat for SDS, which has a molecular weight of 288.4. Thesults are consistent with theories that predict an invelationship between diffusion coefficient and moleceight.The diffusion coefficients for the long-time approximat

an be obtained in a similar approach used for the shortpproximation. Arbitrary diffusion coefficients were sub

uted into Eq. [27] until the calculated curve fit the experimal data near the end of the adsorption process. The resuhe curve fitting for Triton X-100 and SDS are shown in F1 and 12, respectively. The diffusion coefficients obtainedhown in Table 2.

s

FIG. 13. Comparison of the predicted dynamic surface tension (solid5 0.97 3 1026 cm2/s) with the dynamic surface tension measured

rop mass technique for Triton X-100 (h, 0.00078 M;‚, 0.00025 M;E,.00016 M).

FIG. 14. Comparison of the predicted dynamic surface tension (solid5 0.79 3 1026 cm2/s) with the dynamic surface tension measured

rop mass technique for SDS (h, 0.02 M;‚, 0.05 M).

mes tep facb ana lmp sics atem ongt nd mD ams erad at tei wlyg risb e dn iqT tes urw foT

entald gooda cal-c thed ionc rfacet

imea rth oef-fi nts,f thel tenceo achest [47]–[T ergyb (a 8.0R riersa e thel bar-r urityi n oft rrier,a dsorbv m-i hasb 29).

iquec de-s bovet ledm wasp oef-fi que

sc

a-t


The classic techniques for the dynamic surface tensionurement, such as drop mass technique, attempt to simulahysical situation of a newly generated aqueous/air interut have the drawback of obtaining data at the short timeccurately defining the surface age. The dynamic Wilhelate technique is further removed from the actual phyituation of a newly generated interface, but does accureasure the diffusion coefficients for the short- and the l

ime approximations. Since the dynamic surface tensioiffusion controlled, the diffusion coefficients resulting froWPT measurements can be used to predict the dynurface tension for the newly generated interface. The aviffusion coefficients obtained from the long-time approxim

ion for the dynamic Wilhelmy plate technique are substitunto Eq. [4] to predict the long-time approximation for a neenerated aqueous/air interface and thus allow a compaetween the predicted dynamic surface tension and thamic surface tension measured with the drop mass technhe comparison between the predicted dynamic surfaceion (solid line) and the experimental values (points) measith the drop mass technique (12) are shown in Fig. 13riton X-100 and Fig. 14 for SDS.

FIG. 15. Dynamic surface tension vsf(t) for Triton X-100 at variouoncentrations (symbols as in Fig. 7).

FIG. 16. Dynamic surface tension vsf(t) for SDS at various concentrions (symbols as in Fig. 8).

a-thee,dyally-

is

icge-d

ony-ue.n-

edr

The predicted dynamic surface tension and the experimynamic surface tension from drop mass technique are ingreement, which suggests that the diffusion coefficientsulated from the dynamic surface tension measured withynamic Wilhelmy plate technique are the effective diffusoefficients and can be used to predict the dynamic suension for a newly generated surface.

Table 2 shows that the diffusion coefficient for the long-tpproximation for Triton X-100 (0.973 1026 cm2/s) is highe

han that for SDS (0.793 1026 cm2/s), though Triton X-100as a higher molecular weight than SDS. The diffusion ccients for the short-time approximation for both surfactaurthermore, are higher than the diffusion coefficients forong-time approximation. These results suggest the exisf an energy barrier when the aqueous/air interface appro

he saturated adsorption. Treatment of the data with Eqs.49] reveals the linear relationship betweeng and f(t) forriton X-100 and SDS as shown in Figs. 15 and 16. Enarriers calculated from the slope of the linear relationshipK)re in the range of 4.1–5.7 RT for Triton X-100 and 6.5–T for SDS, as shown in Table 3. These adsorption barppear to be a function of surfactant concentration, wher

ower the surfactant concentration, the lower the energyier. This result could be caused by the surface-active impn the surfactant samples, which mimic a slow continuatiohe adsorption process, and simulate an adsorption balthough the main surfactant as well as the contaminant aia diffusion controlled mechanism. Similar effect of contanation in the surfactant sample on the adsorption kineticseen observed and discussed by other researchers (28,

SUMMARY

This study involves the development of a new technalled the dynamic Wilhelmy plate technique, which wasigned based on the concept of introducing a water layer ahe surfactant solution. A theoretical diffusion controlodel for the short-time and long-time approximationsroposed for this experimental technique. The diffusion ccients for Triton X-100 and SDS obtained with this techni

TABLE 3The Calculated Surface Diffusion Coefficients

and the Energy Barrier

Surfactant type Concentration (M) Q (RT)

Triton X-100 0.003 5.70.0006 5.30.00025 5.00.0001 4.1

SDS 0.05 8.00.01 6.90.005 6.5

a i-ma fact ew Ao usi d ee 100a

terds

ant97

11

1

11

1111

1 . V.,

222 r-

2 lev,

2 ew

22 ress,

2 iley-

2

2 tion


re 3.73 1026 and 4.63 1026 cm2/s for short-time approxation and 0.973 1026 and 0.793 1026 cm2/s for long-timepproximation, respectively. The predicted dynamic sur

ension from these diffusion coefficients is in good agreemith the experimental data by the drop mass technique.ther model that considers the energy barrier at the aqueo

nterface was also proposed in this study. The calculatergy barriers are in the range of 4.1–5.7 RT for Triton X-nd 6.5–8.0 RT for SDS.

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Dynamic Surface Tension Measurement with a Dynamic Wilhelmy Plate Technique